CN107391462A

CN107391462A - It is a kind of that string encoding and coding/decoding method are deleted based on entangling for finite state conversion

Info

Publication number: CN107391462A
Application number: CN201710576899.5A
Authority: CN
Inventors: 邵俊; 周永生
Original assignee: Jiangsu Ding Chang Polytron Technologies Inc
Current assignee: Jiangsu Ding Chang Polytron Technologies Inc
Priority date: 2017-07-14
Filing date: 2017-07-14
Publication date: 2017-11-24
Anticipated expiration: 2037-07-14
Also published as: CN107391462B

Abstract

String encoding and coding/decoding method are deleted based on entangling for finite state conversion the invention discloses a kind of, this method generates 20 outer packing coded strings by the optional 10 identity information character strings for being configured to indicate commercial product identification information in 64 limited characters, then coding；As long as destroyed character is no more than the half of string characters quantity, you can by decoding precise restoration outer packing coded string, so as to extract the identity information character string of commodity, so as to reach the purpose checked using the information of tracing to the source of breakage.Method cost of the invention is cheap, anti-damage ability is strong, is easy to spray, is easy to check, can be to up to 1.15 × 10¹⁸Individual different identity information text string generation outer packing coded string, adapts to the characteristics of quick consumer lines scale of construction is huge；End is checked in goods, user need to only input 20 character strings in the equipment for be mounted with this decoding software to retrieve the production of goods, logistics information according to the information of tracing to the source that decoding obtains.

Description

It is a kind of that string encoding and coding/decoding method are deleted based on entangling for finite state conversion

Technical field

The present invention relates to circulation merchandise news traceability system field, specifically one kind can be in outer packager code character Character in string deletes string encoding and solution by what damaged certain amount remained able to recover based on entangling for finite state conversion Code method.

Background technology

The production of retrospect commodity, logistics information are heavy to closing for commodity production business in real time during commodity circulation Will.But because tracing information caused by transportation or human factor is lost, defect causes traceability system can not normal work.

The REID used in traditional traceability system due to needing to use chip as electronic tag, cost compared with Height, also it is unfavorable for installing in commodity outer packing, therefore is not suitable for using in fast consumables industry；Quick Response Code is also to trace back The technology being used widely in the system of source, but the technology fault-tolerance is relatively low, it is easy to lose the identity information of commodity；In recent years It is also popular to carry out a kind of square matrix code scheme based on dot matrix, the program has a higher error correcting capability, but due to currently without Ripe reading equipment, dot matrix, which is directly read, in addition has difficulties, extremely inconvenient in inspection end use.

The content of the invention

The purpose of the present invention is the problem of presence for prior art, there is provided a kind of to be deleted based on entangling for finite state conversion Text string generation and restored method；This method has the advantages that cost is cheap, anti-damage ability is strong, is easy to spray, is easy to inspection, And can be to up to 1.15 × 10¹⁸Individual identity information character string is encoded, and is suitable for the needs in fast-moving consumer goods market；For Wanton destruction mode, when coded string only remaining half in outer packing, this method also can be accurate to commercial product identification information Recover, so as to reach the purpose checked using the information of tracing to the source of breakage.

The purpose of the present invention solves by the following technical programs：

It is a kind of that string encoding and coding/decoding method are deleted based on entangling for finite state conversion, it is characterised in that：This method Step is as follows：

(1) 10 identity information character strings, are distributed for commodity to indicate its identity information；

(2) each character in identity information character string, is mapped as a nonnegative integer for showing its state, and will be non- Negative integer obtains the finite state sequence a that a length is 10 according to the sequencing of character₀,a₁…a₉, make each nonnegative integer be The polynomial each term coefficient of identity information, then identity information multinomial be

(3), construction removes formulaSo that should be except formula be with α⁰,α¹…α⁹For polynomial zero point；

(4) identity information multinomial, is obtained into check polynomial r (X), r (X)=m (X) mod g with removing formula modulo operation (X) check polynomial, is subtracted from identity information multinomial, obtains outer packing coding polynomial C (X)=m (X)-r (X)；

(5) it is, outer packing according to the order inverse mapping of polynomial order from low to high by each term coefficient of outer packing multinomial Coded string；

(6), check end and read outer packing coded string, destruction situation of the reading program according to outer packing coded string There are three kinds of countermeasures：1. without breakage, directly extract identity information character string and obtain the progress commodity examination of its identity information；It is 2. outer The breakage of packager code character string calls core codec module Exact recovery to go out original outer packing coded word within the scope of it can entangle and delete Symbol string, then identity information character string is extracted, carry out commodity examination；3. the destruction character quantity of outer packing coded string is beyond solution The code accurate decoding ranges of module, decoder return to error message code, terminate decoding effort；

(7), such as outer packing coded string, which exists, destroys character and destroys quantity within the scope of accurate decoding, decodes mould Block detects to obtain destroyed character quantity N, is destroyed position β of the character in outer packager code character string₀,β₁…β_N-1, and It is to delete symbol multinomial C'(X by remaining character string maps), the character coefficient of correspondence destroyed is set to 0；

(8) the deletion code polynomial E (X) that destroyed character is formed, is solved, if destroyed character coefficient of correspondence is followed successively by e₀,e₁,e₂…e_N-1, there is deletion code polynomialMay be such that C (X)=C'(X)+E (X), obtained by step (4) C (X) root is α⁰,α¹…α⁹, then C (α^m)=C'(α^m)+E(α^m)=0, m=0,1 ... 9, then E (α^m)=- C'(α^m), will α⁰,α¹…α^N-1Bring E (X) and C'(X) into successively, obtain system of linear equations Ax=b, the j rows i row of wherein matrix A are defined asJ-th of element definition of right-hand vector is b_j=-C'(α^j), x solves line to delete code polynomial unknowm coefficient vector Property equation group, obtain delete code polynomial E (X)；

(9) E (X) is superimposed upon C'(X) on, you can outer packing coding polynomial C (X) is obtained, outer packing is encoded multinomial Formula is put into each term coefficient and deploys to obtain finite state integer sequence, outer packing coded string is obtained after inverse mapping, extraction obtains Merchandise news can be checked after identity information character string.

Described step (1)-(5) are cataloged procedure, and the process can be to 1.15 × 10¹⁸Individual identity information string encoding Generate outer packing coded string.

Outer packing coded string in the step (5) is made up of 20 characters, and wherein top ten character is check word Symbol string, rear ten characters are identity information character string.

Identity information character string in the step (1) by 10 Arabic numerals 0-9,26 English capitalization A-Z, Wantonly ten compositions in 26 English lower case a-z and 2 additional character-#.

The alphabet in outer packing coded string in the step (5) is by 10 Arabic numerals 0-9,26 English 20 compositions in literary capitalization A-Z, 26 English lower case a-z and 2 additional character-#.

Identity information character string in the step (1) is used to check by the identity information character string in traceability system The production of the commodity referred to, circulation information.

The formula exponent number that removes defined in the step (3) is 10, and the multinomial is more using ten different finite state integers Item formula simple zero.

Character string breakage in the step (6) can entangle delete within the scope of refer to impaired outer packing code character The character quantity of string is no more than 10.

Described step (6)-(9) are decoding process, and the decoding process can be to impaired outer packing coded string Outer packing coded string of the character quantity no more than 10 carries out accurate recovery, decoding.

The decoding process of step (6)-(9) only with outer packing coded string to be damaged character quantity relevant.

The present invention has the following advantages compared with prior art：

The present invention method cost is cheap, anti-damage ability is strong, is easy to spray, is easy to check, due to without electronic chip, And outer packing coded string is printed on commodity only with spraying method, cost is greatly reduced；Codified amount is big, can be to more Up to 1.15 × 10¹⁸Individual different identity information text string generation outer packing coded string, adapts to the quick consumer lines scale of construction The characteristics of huge；Anti-damage ability is strong, when destruction commodity outer packing character string, and the character destroyed is no more than word in any way Symbol string character quantity half when, this method can precise restoration outer packing coded string, so as to extract the identity of commodity Information character string, so as to reach the purpose checked using the information of tracing to the source of breakage；It is easy to check, checks end in goods, use Family need to only input the information retrieval of tracing to the source that 20 character strings can obtain according to decoding in the equipment for be mounted with this decoding software Production, logistics information to goods.

Brief description of the drawings

Accompanying drawing 1 is finite state integer-character corresponding table of the invention；

Accompanying drawing 2 is finite state integer-logarithmic table of the invention；

Accompanying drawing 3 is finite state integer-index table of the invention；

Accompanying drawing 4 is coding flow chart of the invention；

Accompanying drawing 5 is commodity outer packing coded string form schematic diagram of the invention；

Accompanying drawing 6 is decoding process figure of the invention.

Embodiment

The present invention is further illustrated with embodiment below in conjunction with the accompanying drawings.

As shown in figures 1 to 6：It is a kind of that string encoding and coding/decoding method are deleted based on entangling for finite state conversion, this method Step is as follows：(1) 10 identity information character strings, are distributed for commodity to indicate its identity information, identity information character string By 10 Arabic numerals 0-9,26 English capitalization A-Z, 26 English lower case a-z and 2 additional character $-# In wantonly ten compositions, according to the identity information string search database, production, the stream communication of the goods can be inquired about Breath；(2) each character in identity information character string, is mapped as a nonnegative integer for showing its state, and will be non-negative whole Number obtains the finite state sequence a that a length is 10 according to the sequencing of character₀,a₁…a₉, make each nonnegative integer be The polynomial each term coefficient of identity information, then identity information multinomial be(3), construction removes formulaSo that should be except formula be with α⁰,α¹…α⁹For polynomial zero point；(4), by identity information multinomial with removing Formula modulo operation, check polynomial r (X), r (X)=m (X) mod g (X) are obtained, it is more that verification is subtracted from identity information multinomial Item formula, obtains outer packing coding polynomial C (X)=m (X)-r (X)；(5), by each term coefficient of outer packing multinomial according to multinomial The order inverse mapping of exponent number from low to high is outer packing coded string, and outer packing coded string is made up of 20 characters, its Middle top ten character is identity information character string for check character string, rear ten characters, specifically outer packing coded string In alphabet by 10 Arabic numerals 0-9,26 English capitalization A-Z, 26 English lower case a-z and 2 20 in individual additional character-# form and are formed by 10 identity information string encodings；(6), check end and read outsourcing Coded string is filled, reading program there are three kinds of countermeasures according to the destruction situation of outer packing coded string：1. without breakage, directly Connect extraction identity information character string and obtain the progress commodity examination of its identity information；2. the breakage of outer packing coded string is deleted can entangle Within the scope of, call core codec module Exact recovery to go out original outer packing coded string, then identity information character string is extracted, Carry out commodity examination；3. the destruction character quantity of outer packing coded string exceeds the accurate decoding ranges of decoder module, decoder Error message code is returned, terminates decoding effort；(7), such as outer packing coded string, which exists, destroys character and destroys quantity in standard Within true decoding ranges, decoder module detects to obtain destroyed character quantity N, is destroyed character in outer packager code character string In position β₀,β₁…β_N-1, and be to delete symbol multinomial C'(X by remaining character string maps), the character destroyed is corresponding Coefficient is set to 0；(8) the deletion code polynomial E (X) that destroyed character is formed, is solved, if destroyed character coefficient of correspondence is successively For e₀,e₁,e₂…e_N-1, there is deletion code polynomialMay be such that C (X)=C'(X)+E (X), by step (4) The root for obtaining C (X) is α⁰,α¹…α⁹, then C (α^m)=C'(α^m)+E(α^m)=0, m=0,1 ... 9, then E (α^m)=- C'(α^m), By α⁰,α¹…α^N-1Bring E (X) and C'(X) into successively, obtain system of linear equations Ax=b, the j rows i row of wherein matrix A are defined asJ-th of element definition of right-hand vector is b_j=-C'(α^j), x solves line to delete code polynomial unknowm coefficient vector Property equation group, obtain delete code polynomial E (X)；(9) E (X) is superimposed upon C'(X) on, you can obtain outer packing coding polynomial C (X), outer packing coding polynomial is put into each term coefficient and deploys to obtain finite state integer sequence, outsourcing is obtained after inverse mapping Coded string is filled, extraction can check merchandise news after obtaining identity information character string.

In the above-mentioned methods, step (1)-(5) are cataloged procedure, and the process can be to 1.15 × 10¹⁸Individual identity information word Accord with string encoding generation outer packing coded string；And the formula exponent number that removes defined in step (3) is 10, the multinomial is with ten differences Finite state integer be multinomial simple zero, can neatly be constructed except formula by suitably choosing zero point.Suddenly (6)-(9) are Decoding process, the decoding process can be no more than the outer packing of 10 to the character quantity of impaired outer packing coded string Coded string carries out accurate recovery, decoding, and the decoding process of step (6)-(9) and the quilt in outer packing coded string It is relevant to damage character quantity；Specifically, the character string breakage in step (6) can entangle delete within the scope of refer to it is impaired The character quantity of outer packing coded string is no more than 10.

Additionally need the rule of the arithmetic to finite state integer for disclosing and being defined when the present invention realizes.With integer Collection is different, and finite state integer represents state limited in system, and its value is only the subset of set of integers, and accompanying drawing 1 is then this Finite state integer-character corresponding table employed in invention, 64 identity information characters are able to count 64 with 0-63 Position nonnegative integer refers to.The computing being defined on finite state integer is positively retained at the closure in the limited manifold, i.e., any The operation result of two finite state integers is still in limited manifold.The add operation being defined in limited manifold is defined as two The step-by-step XOR of number, subtraction are consistent with addition rule.Add operation table 1-1~1-4 is limited to be defined on Addition rule in manifold, the first row and first is put into form and is classified as the integer for participating in computing, due to form is 64 × 64 rule, an add operation table fails to lay down the content, therefore is divided into four parts, below multiplying table and division arithmetic table it is same Because the reason is divided into four parts of compositions.The horizontal nonnegative integer 0-31 of wherein add operation table 1-1 expressions is non-negative whole with longitudinal direction Addition rule between number 0-31；Add operation table 1-2 represents horizontal nonnegative integer 0-31 and longitudinal nonnegative integer Addition rule between 32-63；Add operation table 1-3 represents horizontal nonnegative integer 32-63 and longitudinal nonnegative integer Addition rule between 0-31；Add operation table 1-4 represents horizontal nonnegative integer 32-63 and longitudinal nonnegative integer Addition rule between 32-63.

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31
																																	0	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31
1	1	0	3	2	5	4	7	6	9	8	11	10	13	12	15	14	17	16	19	18	21	20	23	22	25	24	27	26	29	28	31	30
																																	2	2	3	0	1	6	7	4	5	10	11	8	9	14	15	12	13	18	19	16	17	22	23	20	21	26	27	24	25	30	31	28	29
3	3	2	1	0	7	6	5	4	11	10	9	8	15	14	13	12	19	18	17	16	23	22	21	20	27	26	25	24	31	30	29	28
																																	4	4	5	6	7	0	1	2	3	12	13	14	15	8	9	10	11	20	21	22	23	16	17	18	19	28	29	30	31	24	25	26	27
5	5	4	7	6	1	0	3	2	13	12	15	14	9	8	11	10	21	20	23	22	17	16	19	18	29	28	31	30	25	24	27	26
																																	6	6	7	4	5	2	3	0	1	14	15	12	13	10	11	8	9	22	23	20	21	18	19	16	17	30	31	28	29	26	27	24	25
7	7	6	5	4	3	2	1	0	15	14	13	12	11	10	9	8	23	22	21	20	19	18	17	16	31	30	29	28	27	26	25	24
																																	8	8	9	10	11	12	13	14	15	0	1	2	3	4	5	6	7	24	25	26	27	28	29	30	31	16	17	18	19	20	21	22	23
9	9	8	11	10	13	12	15	14	1	0	3	2	5	4	7	6	25	24	27	26	29	28	31	30	17	16	19	18	21	20	23	22
																																	10	10	11	8	9	14	15	12	13	2	3	0	1	6	7	4	5	26	27	24	25	30	31	28	29	18	19	16	17	22	23	20	21
11	11	10	9	8	15	14	13	12	3	2	1	0	7	6	5	4	27	26	25	24	31	30	29	28	19	18	17	16	23	22	21	20
																																	12	12	13	14	15	8	9	10	11	4	5	6	7	0	1	2	3	28	29	30	31	24	25	26	27	20	21	22	23	16	17	18	19
13	13	12	15	14	9	8	11	10	5	4	7	6	1	0	3	2	29	28	31	30	25	24	27	26	21	20	23	22	17	16	19	18
																																	14	14	15	12	13	10	11	8	9	6	7	4	5	2	3	0	1	30	31	28	29	26	27	24	25	22	23	20	21	18	19	16	17
15	15	14	13	12	11	10	9	8	7	6	5	4	3	2	1	0	31	30	29	28	27	26	25	24	23	22	21	20	19	18	17	16
																																	16	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
17	17	16	19	18	21	20	23	22	25	24	27	26	29	28	31	30	1	0	3	2	5	4	7	6	9	8	11	10	13	12	15	14
																																	18	18	19	16	17	22	23	20	21	26	27	24	25	30	31	28	29	2	3	0	1	6	7	4	5	10	11	8	9	14	15	12	13
19	19	18	17	16	23	22	21	20	27	26	25	24	31	30	29	28	3	2	1	0	7	6	5	4	11	10	9	8	15	14	13	12
																																	20	20	21	22	23	16	17	18	19	28	29	30	31	24	25	26	27	4	5	6	7	0	1	2	3	12	13	14	15	8	9	10	11
21	21	20	23	22	17	16	19	18	29	28	31	30	25	24	27	26	5	4	7	6	1	0	3	2	13	12	15	14	9	8	11	10
																																	22	22	23	20	21	18	19	16	17	30	31	28	29	26	27	24	25	6	7	4	5	2	3	0	1	14	15	12	13	10	11	8	9
23	23	22	21	20	19	18	17	16	31	30	29	28	27	26	25	24	7	6	5	4	3	2	1	0	15	14	13	12	11	10	9	8
																																	24	24	25	26	27	28	29	30	31	16	17	18	19	20	21	22	23	8	9	10	11	12	13	14	15	0	1	2	3	4	5	6	7
25	25	24	27	26	29	28	31	30	17	16	19	18	21	20	23	22	9	8	11	10	13	12	15	14	1	0	3	2	5	4	7	6
																																	26	26	27	24	25	30	31	28	29	18	19	16	17	22	23	20	21	10	11	8	9	14	15	12	13	2	3	0	1	6	7	4	5
27	27	26	25	24	31	30	29	28	19	18	17	16	23	22	21	20	11	10	9	8	15	14	13	12	3	2	1	0	7	6	5	4
																																	28	28	29	30	31	24	25	26	27	20	21	22	23	16	17	18	19	12	13	14	15	8	9	10	11	4	5	6	7	0	1	2	3
29	29	28	31	30	25	24	27	26	21	20	23	22	17	16	19	18	13	12	15	14	9	8	11	10	5	4	7	6	1	0	3	2
																																	30	30	31	28	29	26	27	24	25	22	23	20	21	18	19	16	17	14	15	12	13	10	11	8	9	6	7	4	5	2	3	0	1
31	31	30	29	28	27	26	25	24	23	22	21	20	19	18	17	16	15	14	13	12	11	10	9	8	7	6	5	4	3	2	1	0

Table 1-1

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31
																																	32	32	33	34	35	36	37	38	39	40	41	42	43	44	45	46	47	48	49	50	51	52	53	54	55	56	57	58	59	60	61	62	63
33	33	32	35	34	37	36	39	38	41	40	43	42	45	44	47	46	49	48	51	50	53	52	55	54	57	56	59	58	61	60	63	62
																																	34	34	35	32	33	38	39	36	37	42	43	40	41	46	47	44	45	50	51	48	49	54	55	52	53	58	59	56	57	62	63	60	61
35	35	34	33	32	39	38	37	36	43	42	41	40	47	46	45	44	51	50	49	48	55	54	53	52	59	58	57	56	63	62	61	60
																																	36	36	37	38	39	32	33	34	35	44	45	46	47	40	41	42	43	52	53	54	55	48	49	50	51	60	61	62	63	56	57	58	59
37	37	36	39	38	33	32	35	34	45	44	47	46	41	40	43	42	53	52	55	54	49	48	51	50	61	60	63	62	57	56	59	58
																																	38	38	39	36	37	34	35	32	33	46	47	44	45	42	43	40	41	54	55	52	53	50	51	48	49	62	63	60	61	58	59	56	57
39	39	38	37	36	35	34	33	32	47	46	45	44	43	42	41	40	55	54	53	52	51	50	49	48	63	62	61	60	59	58	57	56
																																	40	40	41	42	43	44	45	46	47	32	33	34	35	36	37	38	39	56	57	58	59	60	61	62	63	48	49	50	51	52	53	54	55
41	41	40	43	42	45	44	47	46	33	32	35	34	37	36	39	38	57	56	59	58	61	60	63	62	49	48	51	50	53	52	55	54
																																	42	42	43	40	41	46	47	44	45	34	35	32	33	38	39	36	37	58	59	56	57	62	63	60	61	50	51	48	49	54	55	52	53
43	43	42	41	40	47	46	45	44	35	34	33	32	39	38	37	36	59	58	57	56	63	62	61	60	51	50	49	48	55	54	53	52
																																	44	44	45	46	47	40	41	42	43	36	37	38	39	32	33	34	35	60	61	62	63	56	57	58	59	52	53	54	55	48	49	50	51
45	45	44	47	46	41	40	43	42	37	36	39	38	33	32	35	34	61	60	63	62	57	56	59	58	53	52	55	54	49	48	51	50
																																	46	46	47	44	45	42	43	40	41	38	39	36	37	34	35	32	33	62	63	60	61	58	59	56	57	54	55	52	53	50	51	48	49
47	47	46	45	44	43	42	41	40	39	38	37	36	35	34	33	32	63	62	61	60	59	58	57	56	55	54	53	52	51	50	49	48
																																	48	48	49	50	51	52	53	54	55	56	57	58	59	60	61	62	63	32	33	34	35	36	37	38	39	40	41	42	43	44	45	46	47
49	49	48	51	50	53	52	55	54	57	56	59	58	61	60	63	62	33	32	35	34	37	36	39	38	41	40	43	42	45	44	47	46
																																	50	50	51	48	49	54	55	52	53	58	59	56	57	62	63	60	61	34	35	32	33	38	39	36	37	42	43	40	41	46	47	44	45
51	51	50	49	48	55	54	53	52	59	58	57	56	63	62	61	60	35	34	33	32	39	38	37	36	43	42	41	40	47	46	45	44
																																	52	52	53	54	55	48	49	50	51	60	61	62	63	56	57	58	59	36	37	38	39	32	33	34	35	44	45	46	47	40	41	42	43
53	53	52	55	54	49	48	51	50	61	60	63	62	57	56	59	58	37	36	39	38	33	32	35	34	45	44	47	46	41	40	43	42
																																	54	54	55	52	53	50	51	48	49	62	63	60	61	58	59	56	57	38	39	36	37	34	35	32	33	46	47	44	45	42	43	40	41
55	55	54	53	52	51	50	49	48	63	62	61	60	59	58	57	56	39	38	37	36	35	34	33	32	47	46	45	44	43	42	41	40
																																	56	56	57	58	59	60	61	62	63	48	49	50	51	52	53	54	55	40	41	42	43	44	45	46	47	32	33	34	35	36	37	38	39
57	57	56	59	58	61	60	63	62	49	48	51	50	53	52	55	54	41	40	43	42	45	44	47	46	33	32	35	34	37	36	39	38
																																	58	58	59	56	57	62	63	60	61	50	51	48	49	54	55	52	53	42	43	40	41	46	47	44	45	34	35	32	33	38	39	36	37
59	59	58	57	56	63	62	61	60	51	50	49	48	55	54	53	52	43	42	41	40	47	46	45	44	35	34	33	32	39	38	37	36
																																	60	60	61	62	63	56	57	58	59	52	53	54	55	48	49	50	51	44	45	46	47	40	41	42	43	36	37	38	39	32	33	34	35
61	61	60	63	62	57	56	59	58	53	52	55	54	49	48	51	50	45	44	47	46	41	40	43	42	37	36	39	38	33	32	35	34
																																	62	62	63	60	61	58	59	56	57	54	55	52	53	50	51	48	49	46	47	44	45	42	43	40	41	38	39	36	37	34	35	32	33
63	63	62	61	60	59	58	57	56	55	54	53	52	51	50	49	48	47	46	45	44	43	42	41	40	39	38	37	36	35	34	33	32

Add operation table 1-2

	32	33	34	35	36	37	38	39	40	41	42	43	44	45	46	47	48	49	50	51	52	53	54	55	56	57	58	59	60	61	62	63
																																	0	32	33	34	35	36	37	38	39	40	41	42	43	44	45	46	47	48	49	50	51	52	53	54	55	56	57	58	59	60	61	62	63
1	33	32	35	34	37	36	39	38	41	40	43	42	45	44	47	46	49	48	51	50	53	52	55	54	57	56	59	58	61	60	63	62
																																	2	34	35	32	33	38	39	36	37	42	43	40	41	46	47	44	45	50	51	48	49	54	55	52	53	58	59	56	57	62	63	60	61
3	35	34	33	32	39	38	37	36	43	42	41	40	47	46	45	44	51	50	49	48	55	54	53	52	59	58	57	56	63	62	61	60
																																	4	36	37	38	39	32	33	34	35	44	45	46	47	40	41	42	43	52	53	54	55	48	49	50	51	60	61	62	63	56	57	58	59
5	37	36	39	38	33	32	35	34	45	44	47	46	41	40	43	42	53	52	55	54	49	48	51	50	61	60	63	62	57	56	59	58
																																	6	38	39	36	37	34	35	32	33	46	47	44	45	42	43	40	41	54	55	52	53	50	51	48	49	62	63	60	61	58	59	56	57
7	39	38	37	36	35	34	33	32	47	46	45	44	43	42	41	40	55	54	53	52	51	50	49	48	63	62	61	60	59	58	57	56
																																	8	40	41	42	43	44	45	46	47	32	33	34	35	36	37	38	39	56	57	58	59	60	61	62	63	48	49	50	51	52	53	54	55
9	41	40	43	42	45	44	47	46	33	32	35	34	37	36	39	38	57	56	59	58	61	60	63	62	49	48	51	50	53	52	55	54
																																	10	42	43	40	41	46	47	44	45	34	35	32	33	38	39	36	37	58	59	56	57	62	63	60	61	50	51	48	49	54	55	52	53
11	43	42	41	40	47	46	45	44	35	34	33	32	39	38	37	36	59	58	57	56	63	62	61	60	51	50	49	48	55	54	53	52
																																	12	44	45	46	47	40	41	42	43	36	37	38	39	32	33	34	35	60	61	62	63	56	57	58	59	52	53	54	55	48	49	50	51
13	45	44	47	46	41	40	43	42	37	36	39	38	33	32	35	34	61	60	63	62	57	56	59	58	53	52	55	54	49	48	51	50
																																	14	46	47	44	45	42	43	40	41	38	39	36	37	34	35	32	33	62	63	60	61	58	59	56	57	54	55	52	53	50	51	48	49
15	47	46	45	44	43	42	41	40	39	38	37	36	35	34	33	32	63	62	61	60	59	58	57	56	55	54	53	52	51	50	49	48
																																	16	48	49	50	51	52	53	54	55	56	57	58	59	60	61	62	63	32	33	34	35	36	37	38	39	40	41	42	43	44	45	46	47
17	49	48	51	50	53	52	55	54	57	56	59	58	61	60	63	62	33	32	35	34	37	36	39	38	41	40	43	42	45	44	47	46
																																	18	50	51	48	49	54	55	52	53	58	59	56	57	62	63	60	61	34	35	32	33	38	39	36	37	42	43	40	41	46	47	44	45
19	51	50	49	48	55	54	53	52	59	58	57	56	63	62	61	60	35	34	33	32	39	38	37	36	43	42	41	40	47	46	45	44
																																	20	52	53	54	55	48	49	50	51	60	61	62	63	56	57	58	59	36	37	38	39	32	33	34	35	44	45	46	47	40	41	42	43
21	53	52	55	54	49	48	51	50	61	60	63	62	57	56	59	58	37	36	39	38	33	32	35	34	45	44	47	46	41	40	43	42
																																	22	54	55	52	53	50	51	48	49	62	63	60	61	58	59	56	57	38	39	36	37	34	35	32	33	46	47	44	45	42	43	40	41
23	55	54	53	52	51	50	49	48	63	62	61	60	59	58	57	56	39	38	37	36	35	34	33	32	47	46	45	44	43	42	41	40
																																	24	56	57	58	59	60	61	62	63	48	49	50	51	52	53	54	55	40	41	42	43	44	45	46	47	32	33	34	35	36	37	38	39
25	57	56	59	58	61	60	63	62	49	48	51	50	53	52	55	54	41	40	43	42	45	44	47	46	33	32	35	34	37	36	39	38
																																	26	58	59	56	57	62	63	60	61	50	51	48	49	54	55	52	53	42	43	40	41	46	47	44	45	34	35	32	33	38	39	36	37
27	59	58	57	56	63	62	61	60	51	50	49	48	55	54	53	52	43	42	41	40	47	46	45	44	35	34	33	32	39	38	37	36
																																	28	60	61	62	63	56	57	58	59	52	53	54	55	48	49	50	51	44	45	46	47	40	41	42	43	36	37	38	39	32	33	34	35
29	61	60	63	62	57	56	59	58	53	52	55	54	49	48	51	50	45	44	47	46	41	40	43	42	37	36	39	38	33	32	35	34
																																	30	62	63	60	61	58	59	56	57	54	55	52	53	50	51	48	49	46	47	44	45	42	43	40	41	38	39	36	37	34	35	32	33
31	63	62	61	60	59	58	57	56	55	54	53	52	51	50	49	48	47	46	45	44	43	42	41	40	39	38	37	36	35	34	33	32

Add operation table 1-3

	32	33	34	35	36	37	38	39	40	41	42	43	44	45	46	47	48	49	50	51	52	53	54	55	56	57	58	59	60	61	62	63
																																	32	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31
33	1	0	3	2	5	4	7	6	9	8	11	10	13	12	15	14	17	16	19	18	21	20	23	22	25	24	27	26	29	28	31	30
																																	34	2	3	0	1	6	7	4	5	10	11	8	9	14	15	12	13	18	19	16	17	22	23	20	21	26	27	24	25	30	31	28	29
35	3	2	1	0	7	6	5	4	11	10	9	8	15	14	13	12	19	18	17	16	23	22	21	20	27	26	25	24	31	30	29	28
																																	36	4	5	6	7	0	1	2	3	12	13	14	15	8	9	10	11	20	21	22	23	16	17	18	19	28	29	30	31	24	25	26	27
37	5	4	7	6	1	0	3	2	13	12	15	14	9	8	11	10	21	20	23	22	17	16	19	18	29	28	31	30	25	24	27	26
																																	38	6	7	4	5	2	3	0	1	14	15	12	13	10	11	8	9	22	23	20	21	18	19	16	17	30	31	28	29	26	27	24	25
39	7	6	5	4	3	2	1	0	15	14	13	12	11	10	9	8	23	22	21	20	19	18	17	16	31	30	29	28	27	26	25	24
																																	40	8	9	10	11	12	13	14	15	0	1	2	3	4	5	6	7	24	25	26	27	28	29	30	31	16	17	18	19	20	21	22	23
41	9	8	11	10	13	12	15	14	1	0	3	2	5	4	7	6	25	24	27	26	29	28	31	30	17	16	19	18	21	20	23	22
																																	42	10	11	8	9	14	15	12	13	2	3	0	1	6	7	4	5	26	27	24	25	30	31	28	29	18	19	16	17	22	23	20	21
43	11	10	9	8	15	14	13	12	3	2	1	0	7	6	5	4	27	26	25	24	31	30	29	28	19	18	17	16	23	22	21	20
																																	44	12	13	14	15	8	9	10	11	4	5	6	7	0	1	2	3	28	29	30	31	24	25	26	27	20	21	22	23	16	17	18	19
45	13	12	15	14	9	8	11	10	5	4	7	6	1	0	3	2	29	28	31	30	25	24	27	26	21	20	23	22	17	16	19	18
																																	46	14	15	12	13	10	11	8	9	6	7	4	5	2	3	0	1	30	31	28	29	26	27	24	25	22	23	20	21	18	19	16	17
47	15	14	13	12	11	10	9	8	7	6	5	4	3	2	1	0	31	30	29	28	27	26	25	24	23	22	21	20	19	18	17	16
																																	48	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
49	17	16	19	18	21	20	23	22	25	24	27	26	29	28	31	30	1	0	3	2	5	4	7	6	9	8	11	10	13	12	15	14
																																	50	18	19	16	17	22	23	20	21	26	27	24	25	30	31	28	29	2	3	0	1	6	7	4	5	10	11	8	9	14	15	12	13
51	19	18	17	16	23	22	21	20	27	26	25	24	31	30	29	28	3	2	1	0	7	6	5	4	11	10	9	8	15	14	13	12
																																	52	20	21	22	23	16	17	18	19	28	29	30	31	24	25	26	27	4	5	6	7	0	1	2	3	12	13	14	15	8	9	10	11
53	21	20	23	22	17	16	19	18	29	28	31	30	25	24	27	26	5	4	7	6	1	0	3	2	13	12	15	14	9	8	11	10
																																	54	22	23	20	21	18	19	16	17	30	31	28	29	26	27	24	25	6	7	4	5	2	3	0	1	14	15	12	13	10	11	8	9
55	23	22	21	20	19	18	17	16	31	30	29	28	27	26	25	24	7	6	5	4	3	2	1	0	15	14	13	12	11	10	9	8
																																	56	24	25	26	27	28	29	30	31	16	17	18	19	20	21	22	23	8	9	10	11	12	13	14	15	0	1	2	3	4	5	6	7
57	25	24	27	26	29	28	31	30	17	16	19	18	21	20	23	22	9	8	11	10	13	12	15	14	1	0	3	2	5	4	7	6
																																	58	26	27	24	25	30	31	28	29	18	19	16	17	22	23	20	21	10	11	8	9	14	15	12	13	2	3	0	1	6	7	4	5
59	27	26	25	24	31	30	29	28	19	18	17	16	23	22	21	20	11	10	9	8	15	14	13	12	3	2	1	0	7	6	5	4
																																	60	28	29	30	31	24	25	26	27	20	21	22	23	16	17	18	19	12	13	14	15	8	9	10	11	4	5	6	7	0	1	2	3
61	29	28	31	30	25	24	27	26	21	20	23	22	17	16	19	18	13	12	15	14	9	8	11	10	5	4	7	6	1	0	3	2
																																	62	30	31	28	29	26	27	24	25	22	23	20	21	18	19	16	17	14	15	12	13	10	11	8	9	6	7	4	5	2	3	0	1
63	31	30	29	28	27	26	25	24	23	22	21	20	19	18	17	16	15	14	13	12	11	10	9	8	7	6	5	4	3	2	1	0

Add operation table 1-4

In order to compactly realize multiplication and division arithmetic, it is also necessary to define logarithmic table in limited manifold and index table (such as Shown in Fig. 2, Fig. 3), multiplication and division arithmetic can be compactly completed with looking into table three times and an add operation.

Multiplying table 2-1~2-4 is the multiplication rule being defined in limited manifold, be put into form the first row and First is classified as the integer for participating in computing.The horizontal nonnegative integer 0-31 of wherein multiplying table 2-1 expressions is non-negative whole with longitudinal direction Multiplication rule between number 0-31；Multiplying table 2-2 represents horizontal nonnegative integer 0-31 and longitudinal nonnegative integer Multiplication rule between 32-63；Multiplying table 2-3 represents horizontal nonnegative integer 32-63 and longitudinal nonnegative integer Multiplication rule between 0-31；Multiplying table 2-4 represents horizontal nonnegative integer 32-63 and longitudinal nonnegative integer Multiplication rule between 32-63.

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31
																																	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
1	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31
																																	2	0	2	4	6	8	10	12	14	16	18	20	22	24	26	28	30	32	34	36	38	40	42	44	46	48	50	52	54	56	58	60	62
3	0	3	6	5	12	15	10	9	24	27	30	29	20	23	18	17	48	51	54	53	60	63	58	57	40	43	46	45	36	39	34	33
																																	4	0	4	8	12	16	20	24	28	32	36	40	44	48	52	56	60	3	7	11	15	19	23	27	31	35	39	43	47	51	55	59	63
5	0	5	10	15	20	17	30	27	40	45	34	39	60	57	54	51	19	22	25	28	7	2	13	8	59	62	49	52	47	42	37	32
																																	6	0	6	12	10	24	30	20	18	48	54	60	58	40	46	36	34	35	37	47	41	59	61	55	49	19	21	31	25	11	13	7	1
7	0	7	14	9	28	27	18	21	56	63	54	49	36	35	42	45	51	52	61	58	47	40	33	38	11	12	5	2	23	16	25	30
																																	8	0	8	16	24	32	40	48	56	3	11	19	27	35	43	51	59	6	14	22	30	38	46	54	62	5	13	21	29	37	45	53	61
9	0	9	18	27	36	45	54	63	11	2	25	16	47	38	61	52	22	31	4	13	50	59	32	41	29	20	15	6	57	48	43	34
																																	10	0	10	20	30	40	34	60	54	19	25	7	13	59	49	47	37	38	44	50	56	14	4	26	16	53	63	33	43	29	23	9	3
11	0	11	22	29	44	39	58	49	27	16	13	6	55	60	33	42	54	61	32	43	26	17	12	7	45	38	59	48	1	10	23	28
																																	12	0	12	24	20	48	60	40	36	35	47	59	55	19	31	11	7	5	9	29	17	53	57	45	33	38	42	62	50	22	26	14	2
13	0	13	26	23	52	57	46	35	43	38	49	60	31	18	5	8	21	24	15	2	33	44	59	54	62	51	36	41	10	7	16	29
																																	14	0	14	28	18	56	54	36	42	51	61	47	33	11	5	23	25	37	43	57	55	29	19	1	15	22	24	10	4	46	32	50	60
15	0	15	30	17	60	51	34	45	59	52	37	42	7	8	25	22	53	58	43	36	9	6	23	24	14	1	16	31	50	61	44	35
																																	16	0	16	32	48	3	19	35	51	6	22	38	54	5	21	37	53	12	28	44	60	15	31	47	63	10	26	42	58	9	25	41	57
17	0	17	34	51	7	22	37	52	14	31	44	61	9	24	43	58	28	13	62	47	27	10	57	40	18	3	48	33	21	4	55	38
																																	18	0	18	36	54	11	25	47	61	22	4	50	32	29	15	57	43	44	62	8	26	39	53	3	17	58	40	30	12	49	35	21	7
19	0	19	38	53	15	28	41	58	30	13	56	43	17	2	55	36	60	47	26	9	51	32	21	6	34	49	4	23	45	62	11	24
																																	20	0	20	40	60	19	7	59	47	38	50	14	26	53	33	29	9	15	27	39	51	28	8	52	32	41	61	1	21	58	46	18	6
21	0	21	42	63	23	2	61	40	46	59	4	17	57	44	19	6	31	10	53	32	8	29	34	55	49	36	27	14	38	51	12	25
																																	22	0	22	44	58	27	13	55	33	54	32	26	12	45	59	1	23	47	57	3	21	52	34	24	14	25	15	53	35	2	20	46	56
23	0	23	46	57	31	8	49	38	62	41	16	7	33	54	15	24	63	40	17	6	32	55	14	25	1	22	47	56	30	9	48	39
																																	24	0	24	48	40	35	59	19	11	5	29	53	45	38	62	22	14	10	18	58	34	41	49	25	1	15	23	63	39	44	52	28	4
25	0	25	50	43	39	62	21	12	13	20	63	38	42	51	24	1	26	3	40	49	61	36	15	22	23	14	37	60	48	41	2	27
																																	26	0	26	52	46	43	49	31	5	21	15	33	59	62	36	10	16	42	48	30	4	1	27	53	47	63	37	11	17	20	14	32	58
27	0	27	54	45	47	52	25	2	29	6	43	48	50	41	4	31	58	33	12	23	21	14	35	56	39	60	17	10	8	19	62	37
																																	28	0	28	56	36	51	47	11	23	37	57	29	1	22	10	46	50	9	21	49	45	58	38	2	30	44	48	20	8	31	3	39	59
29	0	29	58	39	55	42	13	16	45	48	23	10	26	7	32	61	25	4	35	62	46	51	20	9	52	41	14	19	3	30	57	36
																																	30	0	30	60	34	59	37	7	25	53	43	9	23	14	16	50	44	41	55	21	11	18	12	46	48	28	2	32	62	39	57	27	5
31	0	31	62	33	63	32	1	30	61	34	3	28	2	29	60	35	57	38	7	24	6	25	56	39	4	27	58	37	59	36	5	26

Multiplying table 2-1

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31
																																	32	0	32	3	35	6	38	5	37	12	44	15	47	10	42	9	41	24	56	27	59	30	62	29	61	20	52	23	55	18	50	17	49
33	0	33	1	32	2	35	3	34	4	37	5	36	6	39	7	38	8	41	9	40	10	43	11	42	12	45	13	44	14	47	15	46
																																	34	0	34	7	37	14	44	9	43	28	62	27	57	18	48	21	55	56	26	63	29	54	20	49	19	36	6	35	1	42	8	45	15
35	0	35	5	38	10	41	15	44	20	55	17	50	30	61	27	56	40	11	45	14	34	1	39	4	60	31	57	26	54	21	51	16
																																	36	0	36	11	47	22	50	29	57	44	8	39	3	58	30	49	21	27	63	16	52	13	41	6	34	55	19	60	24	33	5	42	14
37	0	37	9	44	18	55	27	62	36	1	45	8	54	19	63	26	11	46	2	39	25	60	16	53	47	10	38	3	61	24	52	17
																																	38	0	38	15	41	30	56	17	55	60	26	51	21	34	4	45	11	59	29	52	18	37	3	42	12	7	33	8	46	25	63	22	48
39	0	39	13	42	26	61	23	48	52	19	57	30	46	9	35	4	43	12	38	1	49	22	60	27	31	56	18	53	5	34	8	47
																																	40	0	40	19	59	38	14	53	29	15	39	28	52	41	1	58	18	30	54	13	37	56	16	43	3	17	57	2	42	55	31	36	12
41	0	41	17	56	34	11	51	26	7	46	22	63	37	12	52	29	14	39	31	54	44	5	61	20	9	32	24	49	43	2	58	19
																																	42	0	42	23	61	46	4	57	19	31	53	8	34	49	27	38	12	62	20	41	3	16	58	7	45	33	11	54	28	15	37	24	50
43	0	43	21	62	42	1	63	20	23	60	2	41	61	22	40	3	46	5	59	16	4	47	17	58	57	18	44	7	19	56	6	45
																																	44	0	44	27	55	54	26	45	1	47	3	52	24	25	53	2	46	29	49	6	42	43	7	48	28	50	30	41	5	4	40	31	51
45	0	45	25	52	50	31	43	6	39	10	62	19	21	56	12	33	13	32	20	57	63	18	38	11	42	7	51	30	24	53	1	44
																																	46	0	46	31	49	62	16	33	15	63	17	32	14	1	47	30	48	61	19	34	12	3	45	28	50	2	44	29	51	60	18	35	13
47	0	47	29	50	58	21	39	8	55	24	42	5	13	34	16	63	45	2	48	31	23	56	10	37	26	53	7	40	32	15	61	18
																																	48	0	48	35	19	5	53	38	22	10	58	41	25	15	63	44	28	20	36	55	7	17	33	50	2	30	46	61	13	27	43	56	8
49	0	49	33	16	1	48	32	17	2	51	35	18	3	50	34	19	4	53	37	20	5	52	36	21	6	55	39	22	7	54	38	23
																																	50	0	50	39	21	13	63	42	24	26	40	61	15	23	37	48	2	52	6	19	33	57	11	30	44	46	28	9	59	35	17	4	54
51	0	51	37	22	9	58	44	31	18	33	55	4	27	40	62	13	36	23	1	50	45	30	8	59	54	5	19	32	63	12	26	41
																																	52	0	52	43	31	21	33	62	10	42	30	1	53	63	11	20	32	23	35	60	8	2	54	41	29	61	9	22	34	40	28	3	55
53	0	53	41	28	17	36	56	13	34	23	11	62	51	6	26	47	7	50	46	27	22	35	63	10	37	16	12	57	52	1	29	40
																																	54	0	54	47	25	29	43	50	4	58	12	21	35	39	17	8	62	55	1	24	46	42	28	5	51	13	59	34	20	16	38	63	9
55	0	55	45	26	25	46	52	3	50	5	31	40	43	28	6	49	39	16	10	61	62	9	19	36	21	34	56	15	12	59	33	22
																																	56	0	56	51	11	37	29	22	46	9	49	58	2	44	20	31	39	18	42	33	25	55	15	4	60	27	35	40	16	62	6	13	53
57	0	57	49	8	33	24	16	41	1	56	48	9	32	25	17	40	2	59	51	10	35	26	18	43	3	58	50	11	34	27	19	42
																																	58	0	58	55	13	45	23	26	32	25	35	46	20	52	14	3	57	50	8	5	63	31	37	40	18	43	17	28	38	6	60	49	11
59	0	59	53	14	41	18	28	39	17	42	36	31	56	3	13	54	34	25	23	44	11	48	62	5	51	8	6	61	26	33	47	20
																																	60	0	60	59	7	53	9	14	50	41	21	18	46	28	32	39	27	17	45	42	22	36	24	31	35	56	4	3	63	13	49	54	10
61	0	61	57	4	49	12	8	53	33	28	24	37	16	45	41	20	1	60	56	5	48	13	9	52	32	29	25	36	17	44	40	21
																																	62	0	62	63	1	61	3	2	60	57	7	6	56	4	58	59	5	49	15	14	48	12	50	51	13	8	54	55	9	53	11	10	52
63	0	63	61	2	57	6	4	59	49	14	12	51	8	55	53	10	33	30	28	35	24	39	37	26	16	47	45	18	41	22	20	43

Multiplying table 2-2

	32	33	34	35	36	37	38	39	40	41	42	43	44	45	46	47	48	49	50	51	52	53	54	55	56	57	58	59	60	61	62	63
																																	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
1	32	33	34	35	36	37	38	39	40	41	42	43	44	45	46	47	48	49	50	51	52	53	54	55	56	57	58	59	60	61	62	63
																																	2	3	1	7	5	11	9	15	13	19	17	23	21	27	25	31	29	35	33	39	37	43	41	47	45	51	49	55	53	59	57	63	61
3	35	32	37	38	47	44	41	42	59	56	61	62	55	52	49	50	19	16	21	22	31	28	25	26	11	8	13	14	7	4	1	2
																																	4	6	2	14	10	22	18	30	26	38	34	46	42	54	50	62	58	5	1	13	9	21	17	29	25	37	33	45	41	53	49	61	57
5	38	35	44	41	50	55	56	61	14	11	4	1	26	31	16	21	53	48	63	58	33	36	43	46	29	24	23	18	9	12	3	6
																																	6	5	3	9	15	29	27	17	23	53	51	57	63	45	43	33	39	38	32	42	44	62	56	50	52	22	16	26	28	14	8	2	4
7	37	34	43	44	57	62	55	48	29	26	19	20	1	6	15	8	22	17	24	31	10	13	4	3	46	41	32	39	50	53	60	59
																																	8	12	4	28	20	44	36	60	52	15	7	31	23	47	39	63	55	10	2	26	18	42	34	58	50	9	1	25	17	41	33	57	49
9	44	37	62	55	8	1	26	19	39	46	53	60	3	10	17	24	58	51	40	33	30	23	12	5	49	56	35	42	21	28	7	14
																																	10	15	5	27	17	39	45	51	57	28	22	8	2	52	62	32	42	41	35	61	55	1	11	21	31	58	48	46	36	18	24	6	12
11	47	36	57	50	3	8	21	30	52	63	34	41	24	19	14	5	25	18	15	4	53	62	35	40	2	9	20	31	46	37	56	51
																																	12	10	6	18	30	58	54	34	46	41	37	49	61	25	21	1	13	15	3	23	27	63	51	39	43	44	32	52	56	28	16	4	8
13	42	39	48	61	30	19	4	9	1	12	27	22	53	56	47	34	63	50	37	40	11	6	17	28	20	25	14	3	32	45	58	55
																																	14	9	7	21	27	49	63	45	35	58	52	38	40	2	12	30	16	44	34	48	62	20	26	8	6	31	17	3	13	39	41	59	53
15	41	38	55	56	21	26	11	4	18	29	12	3	46	33	48	63	28	19	2	13	32	47	62	49	39	40	57	54	27	20	5	10
																																	16	24	8	56	40	27	11	59	43	30	14	62	46	29	13	61	45	20	4	52	36	23	7	55	39	18	2	50	34	17	1	49	33
17	56	41	26	11	63	46	29	12	54	39	20	5	49	32	19	2	36	53	6	23	35	50	1	16	42	59	8	25	45	60	15	30
																																	18	27	9	63	45	16	2	52	38	13	31	41	59	6	20	34	48	55	37	19	1	60	46	24	10	33	51	5	23	42	56	14	28
19	59	40	29	14	52	39	18	1	37	54	3	16	42	57	12	31	7	20	33	50	8	27	46	61	25	10	63	44	22	5	48	35
																																	20	30	10	54	34	13	25	37	49	56	44	16	4	43	63	3	23	17	5	57	45	2	22	42	62	55	35	31	11	36	48	12	24
21	62	43	20	1	41	60	3	22	16	5	58	47	7	18	45	56	33	52	11	30	54	35	28	9	15	26	37	48	24	13	50	39
																																	22	29	11	49	39	6	16	42	60	43	61	7	17	48	38	28	10	50	36	30	8	41	63	5	19	4	18	40	62	31	9	51	37
23	61	42	19	4	34	53	12	27	3	20	45	58	28	11	50	37	2	21	44	59	29	10	51	36	60	43	18	5	35	52	13	26
																																	24	20	12	36	60	55	47	7	31	17	9	33	57	50	42	2	26	30	6	46	54	61	37	13	21	27	3	43	51	56	32	8	16
25	52	45	6	31	19	10	33	56	57	32	11	18	30	7	44	53	46	55	28	5	9	16	59	34	35	58	17	8	4	29	54	47
																																	26	23	13	35	57	60	38	8	18	2	24	54	44	41	51	29	7	61	39	9	19	22	12	34	56	40	50	28	6	3	25	55	45
27	55	44	1	26	24	3	46	53	42	49	28	7	5	30	51	40	13	22	59	32	34	57	20	15	16	11	38	61	63	36	9	18
																																	28	18	14	42	54	33	61	25	5	55	43	15	19	4	24	60	32	27	7	35	63	40	52	16	12	62	34	6	26	13	17	53	41
29	50	47	8	21	5	24	63	34	31	2	37	56	40	53	18	15	43	54	17	12	28	1	38	59	6	27	60	33	49	44	11	22
																																	30	17	15	45	51	42	52	22	8	36	58	24	6	31	1	35	61	56	38	4	26	3	29	63	33	13	19	49	47	54	40	10	20
31	49	46	15	16	14	17	48	47	12	19	50	45	51	44	13	18	8	23	54	41	55	40	9	22	53	42	11	20	10	21	52	43

Multiplying table 2-3

	32	33	34	35	36	37	38	39	40	41	42	43	44	45	46	47	48	49	50	51	52	53	54	55	56	57	58	59	60	61	62	63
																																	32	48	16	51	19	54	22	53	21	60	28	63	31	58	26	57	25	40	8	43	11	46	14	45	13	36	4	39	7	34	2	33	1
33	16	49	17	48	18	51	19	50	20	53	21	52	22	55	23	54	24	57	25	56	26	59	27	58	28	61	29	60	30	63	31	62
																																	34	51	17	52	22	61	31	58	24	47	13	40	10	33	3	38	4	11	41	12	46	5	39	2	32	23	53	16	50	25	59	30	60
35	19	48	22	53	25	58	28	63	7	36	2	33	13	46	8	43	59	24	62	29	49	18	52	23	47	12	42	9	37	6	32	3
																																	36	54	18	61	25	32	4	43	15	26	62	17	53	12	40	7	35	45	9	38	2	59	31	48	20	1	37	10	46	23	51	28	56
37	22	51	31	58	4	33	13	40	50	23	59	30	32	5	41	12	29	56	20	49	15	42	6	35	57	28	48	21	43	14	34	7
																																	38	53	19	58	28	43	13	36	2	9	47	6	32	23	49	24	62	14	40	1	39	16	54	31	57	50	20	61	27	44	10	35	5
39	21	50	24	63	15	40	2	37	33	6	44	11	59	28	54	17	62	25	51	20	36	3	41	14	10	45	7	32	16	55	29	58
																																	40	60	20	47	7	26	50	9	33	51	27	32	8	21	61	6	46	34	10	49	25	4	44	23	63	45	5	62	22	11	35	24	48
41	28	53	13	36	62	23	47	6	27	50	10	35	57	16	40	1	18	59	3	42	48	25	33	8	21	60	4	45	55	30	38	15
																																	42	63	21	40	2	17	59	6	44	32	10	55	29	14	36	25	51	1	43	22	60	47	5	56	18	30	52	9	35	48	26	39	13
43	31	52	10	33	53	30	32	11	8	35	29	54	34	9	55	28	49	26	36	15	27	48	14	37	38	13	51	24	12	39	25	50
																																	44	58	22	33	13	12	32	23	59	21	57	14	34	35	15	56	20	39	11	60	16	17	61	10	38	8	36	19	63	62	18	37	9
45	26	55	3	46	40	5	49	28	61	16	36	9	15	34	22	59	23	58	14	35	37	8	60	17	48	29	41	4	2	47	27	54
																																	46	57	23	38	8	7	41	24	54	6	40	25	55	56	22	39	9	4	42	27	53	58	20	37	11	59	21	36	10	5	43	26	52
47	25	54	4	43	35	12	62	17	46	1	51	28	20	59	9	38	52	27	41	6	14	33	19	60	3	44	30	49	57	22	36	11
																																	48	40	24	11	59	45	29	14	62	34	18	1	49	39	23	4	52	60	12	31	47	57	9	26	42	54	6	21	37	51	3	16	32
49	8	57	41	24	9	56	40	25	10	59	43	26	11	58	42	27	12	61	45	28	13	60	44	29	14	63	47	30	15	62	46	31
																																	50	43	25	12	62	38	20	1	51	49	3	22	36	60	14	27	41	31	45	56	10	18	32	53	7	5	55	34	16	8	58	47	29
51	11	56	46	29	2	49	39	20	25	42	60	15	16	35	53	6	47	28	10	57	38	21	3	48	61	14	24	43	52	7	17	34
																																	52	46	26	5	49	59	15	16	36	4	48	47	27	17	37	58	14	57	13	18	38	44	24	7	51	19	39	56	12	6	50	45	25
53	14	59	39	18	31	42	54	3	44	25	5	48	61	8	20	33	9	60	32	21	24	45	49	4	43	30	2	55	58	15	19	38
																																	54	45	27	2	52	48	6	31	41	23	33	56	14	10	60	37	19	26	44	53	3	7	49	40	30	32	22	15	57	61	11	18	36
55	13	58	32	23	20	35	57	14	63	8	18	37	38	17	11	60	42	29	7	48	51	4	30	41	24	47	53	2	1	54	44	27
																																	56	36	28	23	47	1	57	50	10	45	21	30	38	8	48	59	3	54	14	5	61	19	43	32	24	63	7	12	52	26	34	41	17
57	4	61	53	12	37	28	20	45	5	60	52	13	36	29	21	44	6	63	55	14	39	30	22	47	7	62	54	15	38	31	23	46
																																	58	39	29	16	42	10	48	61	7	62	4	9	51	19	41	36	30	21	47	34	24	56	2	15	53	12	54	59	1	33	27	22	44
59	7	60	50	9	46	21	27	32	22	45	35	24	63	4	10	49	37	30	16	43	12	55	57	2	52	15	1	58	29	38	40	19
																																	60	34	30	25	37	23	43	44	16	11	55	48	12	62	2	5	57	51	15	8	52	6	58	61	1	26	38	33	29	47	19	20	40
61	2	63	59	6	51	14	10	55	35	30	26	39	18	47	43	22	3	62	58	7	50	15	11	54	34	31	27	38	19	46	42	23
																																	62	33	31	30	32	28	34	35	29	24	38	39	25	37	27	26	36	16	46	47	17	45	19	18	44	41	23	22	40	20	42	43	21
63	1	62	60	3	56	7	5	58	48	15	13	50	9	54	52	11	32	31	29	34	25	38	36	27	17	46	44	19	40	23	21	42

Multiplying table 2-4

Division arithmetic table 3-1~3-4 is the division arithmetic rule being defined in limited manifold, be put into form the first row and First is classified as the integer for participating in computing.The horizontal nonnegative integer 0-31 of wherein division arithmetic table 3-1 expressions is non-negative whole with longitudinal direction Division arithmetic rule between number 0-31；Division arithmetic table 3-2 represents horizontal nonnegative integer 0-31 and longitudinal nonnegative integer Division arithmetic rule between 32-63；Division arithmetic table 3-3 represents horizontal nonnegative integer 32-63 and longitudinal nonnegative integer Division arithmetic rule between 0-31；Division arithmetic table 3-4 represents horizontal nonnegative integer 32-63 and longitudinal nonnegative integer Division arithmetic rule between 32-63.

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31
																																1	1	33	62	49	43	31	44	57	37	52	28	46	40	22	25	61	54	51	39	26	35	14	24	23	15	20	34	11	53	45	6
2	2	1	63	33	21	62	27	49	9	43	56	31	19	44	50	57	47	37	13	52	5	28	48	46	30	40	7	22	41	25	12
																																3	3	32	1	16	62	33	55	8	44	31	36	49	59	58	43	4	25	22	42	46	38	18	40	57	17	60	37	29	28	52	10
4	4	2	61	1	42	63	54	33	18	21	51	62	38	27	39	49	29	9	26	43	10	56	35	31	60	19	14	44	17	50	24
																																5	5	35	3	48	1	32	26	24	55	33	47	16	14	13	62	12	43	58	61	49	41	54	59	8	51	7	44	39	36	31	30
6	6	3	2	32	63	1	45	16	27	62	11	33	53	55	21	8	50	44	23	31	15	36	19	49	34	59	9	58	56	43	20
																																7	7	34	60	17	20	30	1	41	62	10	23	15	29	33	12	53	4	31	48	5	44	42	11	38	45	47	43	49	13	6	18
8	8	4	57	2	23	61	47	1	36	42	37	63	15	54	13	33	58	18	52	21	20	51	5	62	59	38	28	27	34	39	48
																																9	9	37	7	51	60	34	3	56	1	30	57	17	39	32	20	28	12	33	19	15	55	61	29	41	52	50	62	16	23	10	54
10	10	5	6	35	2	3	52	48	45	1	29	32	28	26	63	24	21	55	57	33	17	47	53	16	37	14	27	13	11	62	60
																																11	11	36	56	18	41	28	24	9	8	53	1	14	52	12	38	37	35	4	30	59	50	33	45	7	42	26	57	6	62	19	58
12	12	6	4	3	61	2	25	32	54	63	22	1	41	45	42	16	39	27	46	62	30	11	38	33	7	53	18	55	51	21	40
																																13	13	39	58	50	22	29	53	25	19	11	10	47	1	59	51	45	17	40	9	36	61	5	62	54	8	33	48	60	6	56	46
14	14	7	59	34	40	60	2	17	63	20	46	30	58	1	24	41	8	62	35	10	27	23	22	15	25	29	21	33	26	12	36
																																15	15	38	5	19	3	35	46	40	26	32	50	48	18	23	1	20	62	13	4	16	56	25	14	24	22	9	55	42	47	33	34
16	16	8	49	4	46	57	29	2	11	23	9	61	30	47	26	1	55	36	43	42	40	37	10	63	53	15	56	54	7	13	35
																																17	17	41	15	53	5	38	49	59	46	35	21	19	54	57	3	60	1	23	12	48	11	43	18	40	58	27	26	61	50	32	37
18	18	9	14	37	59	7	6	51	2	60	49	34	13	3	40	56	24	1	38	30	45	57	58	17	43	39	63	32	46	20	47
																																19	19	40	48	20	16	24	42	10	39	8	45	12	37	21	49	5	46	50	1	4	14	55	34	6	36	51	29	43	27	57	41
20	20	10	12	5	4	6	43	35	25	2	58	3	56	52	61	48	42	45	49	1	34	29	41	32	9	28	54	26	22	63	59
																																21	21	43	50	52	47	25	7	26	60	54	38	45	16	34	36	13	28	30	22	27	1	19	49	55	6	8	20	17	35	18	61
22	22	11	51	36	17	56	48	18	16	41	2	28	43	24	15	9	5	8	60	53	39	1	25	14	23	52	49	12	63	38	55
																																23	23	42	13	21	58	39	28	43	53	29	30	50	3	14	22	52	51	59	27	47	4	15	1	25	24	32	19	7	10	11	49
24	24	12	8	6	57	4	50	3	47	61	44	2	17	25	23	32	13	54	31	63	60	22	15	1	14	41	36	45	37	42	19
																																25	25	45	54	55	18	27	30	58	10	9	48	44	57	15	14	29	59	5	56	37	31	24	23	22	1	61	6	38	16	7	21
26	26	13	55	39	44	58	41	50	38	22	20	29	2	53	37	25	34	19	18	11	57	10	63	47	16	1	35	59	12	51	31
																																27	27	44	9	22	7	37	5	11	3	34	8	51	42	35	60	36	20	32	53	17	26	4	39	56	31	21	1	48	57	30	25
28	28	14	53	7	19	59	4	34	61	40	31	60	55	2	48	17	16	63	5	20	54	46	44	30	50	58	42	1	52	24	11
																																29	29	47	11	54	56	36	40	27	24	28	3	18	31	20	41	44	38	12	34	14	21	32	52	9	61	46	8	10	1	53	13
30	30	15	10	38	6	5	31	19	52	3	39	35	36	46	2	40	63	26	8	32	51	50	28	48	44	18	45	23	29	1	7
																																31	31	46	52	23	45	26	51	42	17	55	59	13	12	56	27	21	9	41	47	58	16	60	4	39	35	6	15	28	40	44	1

Division arithmetic table 3-1

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31
																																32	32	16	33	8	31	49	58	4	22	46	18	57	60	29	52	2	45	11	21	23	19	9	20	61	41	30	51	47	14	26	5
33	33	49	31	57	52	46	22	61	51	26	14	23	20	11	45	63	27	56	50	13	48	7	12	42	38	10	17	36	59	55	3
																																34	34	17	30	41	10	15	33	53	31	5	42	38	47	49	6	59	2	46	24	35	22	21	36	19	55	54	52	57	39	3	9
35	35	48	32	24	33	16	13	12	58	49	54	8	7	39	31	6	52	29	63	57	53	27	60	4	56	34	22	50	18	46	15
																																36	36	18	28	9	53	14	12	37	4	59	33	7	26	6	19	51	48	2	15	60	25	49	55	34	21	13	61	3	31	40	29
37	37	51	34	56	30	17	32	28	33	15	61	41	50	16	10	14	6	49	40	38	58	63	47	53	26	25	31	8	42	5	27
																																38	38	19	35	40	32	48	23	20	13	16	25	24	9	42	33	10	31	39	2	8	28	45	7	12	11	37	58	21	54	49	17
39	39	50	29	25	11	47	59	45	40	36	5	54	33	60	56	55	41	20	37	18	63	35	31	27	4	49	24	30	3	28	23
																																40	40	20	24	10	8	12	21	5	50	4	55	6	51	43	57	35	23	25	33	2	7	58	17	3	18	56	47	52	44	61	53
41	41	53	38	59	35	19	57	60	23	48	43	40	27	61	32	30	33	42	6	24	36	52	9	20	29	44	13	63	25	16	51
																																42	42	21	39	43	29	50	14	52	59	47	15	25	32	7	11	26	56	60	44	54	2	38	33	45	12	16	40	34	5	36	57
43	43	52	25	26	54	45	34	13	30	27	19	55	8	17	18	39	14	15	11	44	33	40	57	58	3	4	10	41	48	9	63
																																44	44	22	37	11	34	51	35	36	32	17	4	56	21	48	30	18	10	16	59	41	13	2	50	28	46	43	33	24	61	15	45
45	45	55	27	58	9	44	15	29	5	37	24	22	61	38	7	47	60	35	28	51	46	12	42	11	33	63	3	19	8	34	43
																																46	46	23	26	42	55	13	56	21	41	58	60	39	6	28	44	43	37	53	54	29	8	30	2	50	48	3	38	14	20	22	33
47	47	54	36	27	28	18	20	44	12	14	32	9	46	10	53	22	19	6	17	7	43	16	26	37	63	23	4	5	33	59	39
																																48	48	24	16	12	49	8	39	6	29	57	27	4	34	50	46	3	26	47	62	61	59	44	30	2	28	17	11	25	9	23	38
49	49	57	46	61	26	23	11	63	56	13	7	42	10	36	55	62	44	28	25	39	24	34	6	21	19	5	41	18	60	58	32
																																50	50	25	47	45	36	54	60	55	20	18	35	27	49	30	28	58	53	10	51	9	62	48	46	44	2	57	12	15	32	14	42
51	51	56	17	28	15	41	16	14	49	38	63	53	25	8	5	7	3	57	20	19	29	62	54	59	13	45	46	4	21	35	44
																																52	52	26	45	13	27	55	17	39	15	44	40	58	4	41	9	50	7	38	36	22	49	20	61	29	32	2	5	53	24	37	62
53	53	59	19	60	48	40	61	30	42	24	52	20	44	63	16	15	49	21	3	12	18	26	37	10	47	22	39	62	45	8	56
																																54	54	27	18	44	14	9	10	22	6	7	16	37	23	5	59	11	40	3	41	34	52	8	13	51	62	42	2	35	49	60	50
55	55	58	44	29	37	22	38	47	35	51	12	11	63	19	34	54	30	48	14	56	23	6	21	36	49	62	32	40	4	17	52
																																56	56	28	41	14	38	53	8	7	57	19	62	59	45	4	35	34	32	61	10	40	47	31	27	60	39	55	23	2	43	48	22
57	57	61	23	63	13	42	36	62	28	39	34	21	5	18	58	31	22	14	45	50	12	17	3	43	40	35	53	9	30	29	16
																																58	58	29	22	47	51	11	19	54	48	56	6	36	62	40	17	27	15	24	7	28	42	3	43	18	57	31	16	20	2	41	26
59	59	60	40	30	24	20	63	15	21	12	26	10	22	62	8	38	57	43	32	6	9	13	51	5	54	11	50	31	55	4	28
																																60	60	30	20	15	12	10	62	38	43	6	13	5	11	31	4	19	61	52	16	3	37	39	56	35	27	36	25	46	58	2	14
61	61	63	42	62	39	21	18	31	14	50	17	43	35	9	29	46	11	7	55	25	6	41	32	52	20	48	59	37	15	47	8
																																62	62	31	43	46	25	52	37	23	34	45	53	26	24	51	54	42	18	17	29	55	32	59	8	13	5	12	30	56	19	27	2
63	63	62	21	31	50	43	9	46	7	25	41	52	48	37	47	23	36	34	58	45	3	53	16	26	10	24	60	51	38	54	4

Division arithmetic table 3-2

	32	33	34	35	36	37	38	39	40	41	42	43	44	45	46	47	48	49	50	51	52	53	54	55	56	57	58	59	60	61	62	63
																																	1	63	2	27	21	56	9	50	19	13	47	48	5	7	30	12	41	42	4	38	18	10	29	17	60	36	8	59	58	55	16	3	32
2	61	4	54	42	51	18	39	38	26	29	35	10	14	60	24	17	23	8	15	36	20	58	34	59	11	16	53	55	45	32	6	3
																																	3	2	6	45	63	11	27	21	53	23	50	19	15	9	34	20	56	61	12	41	54	30	39	51	7	47	24	14	13	26	48	5	35
4	57	8	47	23	37	36	13	15	52	58	5	20	28	59	48	34	46	16	30	11	40	55	7	53	22	32	41	45	25	3	12	6
																																	5	6	10	52	2	29	45	63	28	57	21	53	17	27	37	60	11	4	20	56	25	34	42	22	9	50	40	18	23	46	19	15	38
6	4	12	25	61	22	54	42	41	46	39	38	30	18	7	40	51	57	24	17	47	60	13	37	14	29	48	28	26	52	35	10	5
																																	7	59	14	2	40	46	63	24	58	35	8	22	27	21	25	36	26	19	28	55	61	54	16	52	50	57	56	39	32	3	51	9	37
8	49	16	29	46	9	11	26	30	43	55	10	40	56	53	35	7	31	32	60	22	19	45	14	41	44	3	17	25	50	6	24	12
																																	9	14	18	6	59	49	2	40	13	38	24	58	45	63	43	47	46	53	36	26	4	25	48	31	21	8	11	42	35	5	22	27	44
10	12	20	43	4	58	25	61	56	49	42	41	34	54	9	59	22	8	40	51	50	7	23	44	18	39	19	36	46	31	38	30	15
																																	11	51	22	48	17	2	16	15	43	60	5	25	39	49	23	55	63	34	44	21	32	13	10	61	46	3	27	31	20	40	54	29	47
12	8	24	50	57	44	47	23	17	31	13	15	60	36	14	19	37	49	48	34	29	59	26	9	28	58	35	56	52	43	5	20	10
																																	13	55	26	41	44	20	38	37	2	18	34	63	57	35	16	31	12	27	52	4	15	49	7	24	32	30	43	3	14	28	21	23	42
14	53	28	4	19	31	61	48	55	5	16	44	54	42	50	11	52	38	56	45	57	47	32	43	39	49	51	13	3	6	37	18	9
																																	15	10	30	31	6	39	52	2	36	8	63	28	51	45	44	7	29	12	60	11	43	37	61	58	27	21	59	54	57	49	53	17	41
16	33	32	58	31	18	22	52	60	21	45	20	19	51	41	5	14	62	3	59	44	38	25	28	17	27	6	34	50	39	12	48	24
																																	17	30	34	33	10	42	31	6	47	24	2	36	22	52	55	9	39	20	7	29	62	44	4	13	45	63	14	25	8	16	28	51	56
18	28	36	12	53	33	4	19	26	15	48	55	25	61	21	29	31	41	11	52	8	50	35	62	42	16	22	23	5	10	44	54	27
																																	19	35	38	23	32	25	13	33	9	2	31	7	28	58	11	17	54	3	15	18	26	56	62	47	22	52	30	44	63	61	60	53	59
20	24	40	21	8	55	50	57	51	33	23	17	7	47	18	53	44	16	19	37	39	14	46	27	36	13	38	11	31	62	15	60	30
																																	21	39	42	14	29	15	59	11	32	44	56	33	2	40	12	57	5	58	23	3	53	4	51	10	24	41	46	48	37	9	31	63	62
22	37	44	35	34	4	32	30	21	59	10	50	13	33	46	45	61	7	27	42	3	26	20	57	31	6	54	62	40	19	47	58	29
																																	23	26	46	56	55	60	41	44	6	54	37	2	8	38	48	33	20	45	31	12	17	16	9	40	35	34	62	5	18	36	63	57	61
24	16	48	39	49	27	29	46	34	62	26	30	59	11	28	38	9	33	35	7	58	53	52	18	56	55	5	51	43	21	10	40	20
																																	25	47	50	60	36	35	20	28	49	51	53	46	62	12	2	42	32	11	39	33	40	63	41	3	4	19	13	8	17	34	26	43	52
26	45	52	17	27	40	15	9	4	36	7	61	49	5	32	62	24	54	43	8	30	33	14	48	3	60	21	6	28	56	42	46	23
																																	27	18	54	10	14	16	6	59	23	41	40	13	52	2	62	50	49	28	47	46	12	43	19	33	63	24	29	61	38	15	58	45	55
28	41	56	8	38	62	57	35	45	10	32	27	47	23	39	22	43	15	51	25	49	29	3	21	13	33	37	26	6	12	9	36	18
																																	29	22	58	19	51	6	48	17	62	7	15	43	42	16	57	26	2	37	55	63	35	23	30	4	49	5	45	33	60	59	25	39	50
30	20	60	62	12	13	43	4	11	16	61	56	37	25	27	14	58	24	59	22	21	9	57	55	54	42	53	47	49	33	41	34	17
																																	31	43	62	37	25	53	34	54	24	29	18	8	32	30	5	2	19	50	63	48	7	3	36	38	10	14	61	20	11	22	57	33	49

Division arithmetic table 3-3

	32	33	34	35	36	37	38	39	40	41	42	43	44	45	46	47	48	49	50	51	52	53	54	55	56	57	58	59	60	61	62	63
																																	32	1	3	55	62	36	44	43	59	42	25	40	38	37	17	10	28	63	6	53	27	15	50	56	34	54	12	7	39	13	24	35	48
33	62	1	44	43	28	37	25	40	39	54	24	35	34	15	6	53	21	2	19	9	5	47	41	30	18	4	60	29	58	8	32	16
																																	34	60	7	1	20	23	62	12	29	48	4	11	44	43	45	18	13	40	14	58	63	27	8	26	25	61	28	50	16	32	56	37	51
35	3	5	26	1	47	55	62	14	61	43	59	41	44	51	30	36	2	10	28	45	17	21	11	37	25	20	9	42	23	40	38	19
																																	36	56	11	24	41	1	8	38	52	30	35	45	50	57	42	58	62	17	22	43	16	39	5	63	23	32	44	46	10	20	27	47	54
37	7	9	3	60	57	1	20	39	19	12	29	55	62	52	54	23	59	18	13	2	45	24	46	43	4	36	21	48	35	11	44	22
																																	38	5	15	46	3	50	26	1	18	4	62	14	56	55	22	34	47	6	30	36	52	51	63	29	44	43	60	27	61	57	59	41	53
39	58	13	53	22	10	19	51	1	9	17	62	61	48	8	46	6	44	26	2	38	57	34	12	16	15	52	32	7	14	43	42	21
																																	40	48	19	42	16	45	39	49	37	1	46	34	14	29	36	41	27	32	38	9	13	28	31	54	11	26	15	22	62	63	30	59	60
41	15	17	49	5	21	46	3	54	12	1	18	11	26	58	37	50	10	34	47	31	22	2	39	55	62	7	45	4	8	14	56	28
																																	42	13	23	28	58	30	53	22	3	27	51	1	4	19	24	49	10	55	46	6	41	8	37	20	48	17	31	35	9	18	62	61	63
43	50	21	7	47	38	60	36	16	22	28	49	1	20	6	61	35	29	42	32	59	2	56	5	12	53	23	24	51	37	46	62	31
																																	44	9	27	5	7	8	3	60	42	53	20	39	26	1	31	25	57	14	54	23	6	52	40	49	62	12	47	63	19	38	29	55	58
45	54	25	30	18	48	10	14	57	56	59	23	31	6	1	21	16	36	50	49	20	62	53	32	2	40	39	4	41	17	13	52	26
																																	46	52	31	51	45	59	17	27	12	47	9	4	16	15	35	1	40	25	62	24	34	32	18	19	5	7	63	10	36	11	61	49	57
47	11	29	40	56	3	24	41	31	34	38	52	21	8	61	13	1	51	58	62	48	42	15	2	57	35	55	49	30	60	45	50	25
																																	48	32	35	13	33	54	58	31	7	63	52	60	53	22	56	15	18	1	5	14	55	41	43	36	51	45	10	37	21	42	20	19	40
49	31	33	22	52	14	51	45	20	50	27	12	48	17	38	3	59	43	1	40	37	35	54	53	15	9	2	30	47	29	4	16	8
																																	50	29	39	59	11	5	40	56	33	37	41	31	63	24	4	23	3	22	13	1	19	61	17	6	8	38	26	16	34	7	52	21	43
51	34	37	32	30	61	33	10	50	40	6	47	58	31	26	27	42	60	9	39	1	55	12	23	52	2	18	43	24	48	36	22	11
																																	52	25	43	34	54	19	30	18	8	11	14	57	33	10	3	63	48	47	21	16	60	1	28	35	6	59	42	12	56	51	23	31	46
53	38	41	57	35	43	23	32	27	6	33	9	36	13	29	51	25	5	17	54	46	11	1	50	58	31	34	55	2	4	7	28	14
																																	54	36	47	20	28	32	12	53	46	17	19	26	43	4	63	39	33	56	29	31	24	21	38	1	61	48	58	57	15	30	55	25	45
55	27	45	15	9	24	5	7	61	28	60	42	46	3	33	43	8	18	25	57	10	31	59	16	1	20	50	2	53	41	39	26	13
																																	56	17	51	16	15	63	49	5	25	20	3	54	29	46	13	44	21	30	37	50	33	58	6	42	26	1	9	52	12	24	18	11	36
57	46	49	11	26	7	56	55	10	25	44	6	24	41	19	32	60	52	33	20	51	48	27	59	38	37	1	15	54	47	2	8	4
																																	58	44	55	38	37	12	35	34	63	14	30	21	23	32	49	52	4	9	45	61	5	46	60	8	33	10	25	1	59	53	50	13	39
59	19	53	61	48	52	42	16	44	3	49	37	18	39	47	56	45	35	41	27	23	36	33	25	29	46	17	58	1	2	34	14	7
																																	60	40	59	63	24	26	21	8	22	32	57	51	9	50	54	28	55	48	53	44	42	18	49	45	47	23	41	29	33	1	17	7	34
61	23	57	36	13	34	28	58	5	45	22	3	12	53	40	16	30	26	49	10	56	24	44	60	19	51	33	38	27	54	1	4	2
																																	62	21	63	9	50	41	7	47	48	58	36	16	3	60	10	4	38	39	61	35	14	6	11	15	20	28	57	40	22	44	49	1	33
63	42	61	18	39	17	14	29	35	55	11	32	6	59	20	8	15	13	57	5	28	12	22	30	40	56	49	19	44	27	33	2	1

Division arithmetic table 3-4

Below by taking the computing of finite state integer 9 and 16,10 and 12 as an example, according to above-mentioned add operation table 1-1~1- 4th, multiplying table 2-1~2-4 and division arithmetic table 3-1~3-4 illustrates add operation table, multiplying table and division arithmetic The application method of table：

Addition：9+16=25,10+12=6；

Multiplication：9 × 16=22,10 × 12=59；

Division：9/16=28,10/12=32.

Such as when calculating 9+16, by looking into add operation table 1-1, in the corresponding first row of first operand 9 numeral 9 that A line, second operand 16 correspond to that row of numeral 16 in the first row, and the numerical value of the intersection of above-mentioned ranks is 9+16 Operation result 25, other result of calculations obtain also through this method.

Illustrate that the present invention is defined on the exponent arithmetic gexp of finite state integer and logarithm is transported again by example below Calculate glog.So there is equation below when computing a × b and a/b：A × b=gexp [glog (a)+glog (b)] and a/b= gexp[glog(a)-glog(b)].A=9, b=16 are taken, by looking into the logarithmic table of accompanying drawing 2 and the index table of accompanying drawing 3,9 can be obtained × 16=gexp [glog (9)+glog (16)]=gexp [32+4]=gexp [36]=22, and 9/16=gexp [glog (9)- Glog (16)]=gexp [32-4]=gexp [28]=28.

With reference to the accompanying drawings and examples come further illustrate the present invention coding and decoding step, be advantageous to more preferable geography The solution present invention.

Embodiment one：Assuming that the identity information string encoding of certain merchandise is bT6pzDRMzC.

First, specific coding step (as shown in Figure 4)

(1), for commodity distribute 10 identity information character strings for indicating its identity information, in this example for bT6pzDRMzC；

(2) each character of identity information character string, is mapped as a nonnegative integer for showing its state, mapping relations are such as Shown in Fig. 1, according to the sequencing of character obtain a length be 10 finite state sequence 37,29,6,51,61,13,27, 22nd, 61,12, it is the polynomial each term coefficient of identity information to make each integer of finite state sequence, m (X)=37X¹⁰+29X¹¹+6X¹²+ 51X¹³+61X¹⁴+13X¹⁵+27X¹⁶+22X¹⁷+61X¹⁸+12X¹⁹；

(3), definition removes formula, with 1,2,4,8,16,32,3,6,12,24 for polynomial zero point, makes g (X)=25X⁰+28X¹ +42X²+33X³+63X⁴+46X⁵+26X⁶+45X⁷+7X⁸+46X⁹+X¹⁰It is 0 in above-mentioned zero point, in order to illustrate finite state integer Computation rule, be described further below exemplified by g when X=8 (X) computational methods：If the logarithm fortune of finite state integer Calculate and exponent arithmetic is represented by glog, gexp, as shown in Figures 2 and 3, a is multiplied with b is represented by a × b=to its operation relation Gexp { [glog (a)+glog (b)] mod63 }, then X n power Xⁿ=gexp [glog (X) × n], such as 8⁵=gexp [glog (8) × 5]=gexp (3 × 5)=40, can so there is 8⁰~8¹⁰Successively equal to 1,8,3,24,5,40,15,59,17,14,51, then Each monomial is multiplied by each term coefficient successively, such as 46 × 8⁵=46 × 40=gexp { [glog (46)+glog (40)] mod63 }= Gexp (7)=6, can similarly there is every product to be followed successively by 25,37,61,12,6,6,16,4,52,30,51, it is whole according to finite state For number addition tables by above-mentioned product accumulation, its value is 0, can demonstrate,prove that value is all on any polynomial zeros except formula after the same method For 0；

(4) identity information multinomial, is obtained into check polynomial, r (X)=44X with removing formula modulo operation⁰+60X¹+58X² +23X³+38X⁴+59X⁵+49X⁶+45X⁷+X⁸+60X⁹, check polynomial is subtracted from identity information multinomial, obtains outer packing volume Code polynomial is C (X)=44X⁰+60X¹+58X²+23X³+38X⁴+59X⁵+49X⁶+45X⁷+X⁸+60X⁹+37X¹⁰+29X¹¹+6X¹²+ 51X¹³+61X¹⁴+13X¹⁵+27X¹⁶+22X¹⁷+61X¹⁸+12X¹⁹；

(5) it is, outer by order inverse mapping of each term coefficient of outer packing coding polynomial according to polynomial order from low to high Packager code character string, it is iywNcxnj1ybT6pzDRMzC that can obtain outer packing coded string according to Fig. 1 mapping relations, outside Check character substring and identity information character substring form be as shown in Figure 5 in packager code character string.

2nd, specific decoding step (as shown in Figure 6)

(6), check end and read outer packing character string, in character string because damaged and the character of None- identified is accorded with underscore Number _ represent, in embodiment one, the information of outer packing coded string residual is iy___xnj1____pz_RM__；

(7) it is 10 that, decoder module, which detects to obtain destroyed character quantity, is not above entangling for decoder module and deletes model Enclose, destroyed position of the character in outer packager code character string is followed successively by the 2nd, 3,4,9,10,11,12,15,18,19 word Accord with (counting since 0), be then mapped as deleting symbol multinomial C'(X by remaining outer packing coded string)=44X⁰+60X¹+ 59X⁵+49X⁶+45X⁷+X⁸+51X¹³+61X¹⁴+27X¹⁶+22X¹⁷, the character coefficient of correspondence destroyed is set to 0；

(8) the deletion code polynomial E (X) that destroyed character is formed, is solved, it is assumed that the following E of the polynomial that deletes code (X) =e₀X²+e₁X³+e₂X⁴+e₃X⁹+e₄X¹⁰+e₅X¹¹+e₆X¹²+e₇X¹⁵+e₈X¹⁸+e₉X¹⁹, wherein e₀,e₁,e₂…e₉Represent what is be destroyed Character coefficient of correspondence, E (X) and C'(X) will be brought into successively except the zero point of formula, obtain system of linear equations Ax=b, wherein matrix A j Row i row are defined asα^jRepresent j-th of zero point, β_iRepresent the position of i-th of destruction character, j-th yuan of right-hand vector Element is defined as b_j=-C'(α^j), x solves system of linear equations to delete code polynomial unknowm coefficient vector, obtains deleting code multinomial Formula E (X), it is in this example E (X)=58X²+23X³+38X⁴+60X⁹+37X¹⁰+29X¹¹+6X¹²+13X¹⁵+61X¹⁸+12X¹⁹；

(9) E (X) is superimposed upon C'(X) on, you can obtain outer packing coding polynomial C (X), C (X)=44X⁰+60X¹+ 58X²+23X³+38X⁴+59X⁵+49X⁶+45X⁷+X⁸+60X⁹+37X¹⁰+29X¹¹+6X¹²+51X¹³+61X¹⁴+13X¹⁵+27X¹⁶+22X¹⁷+ 61X¹⁸+12X¹⁹, each term coefficient of multinomial is deployed to obtain finite state integer sequence, outer packing coded word is obtained after inverse mapping Symbol string iywNcxnj1ybT6pzDRMzC, extraction can check merchandise news after obtaining identity information character string bT6pzDRMzC.

Embodiment two：Assuming that the identity information string encoding of certain merchandise is isBn978704.

First, specific coding step (as shown in Figure 4)

(1), for commodity distribute 10 identity information character strings for indicating its identity information, in this example for isBn978704；

(2) each character of identity information character string, is mapped as a nonnegative integer for showing its state, mapping relations are such as Shown in Fig. 1, according to the sequencing of character obtain a length be 10 finite state sequence 44,54,11,49,9,7,8,7, 0th, 4, each integer for making finite state sequence is the polynomial each term coefficient of identity information, m (X)=44X¹⁰+54X¹¹+11X¹²+ 49X¹³+9X¹⁴+7X¹⁵+8X¹⁶+7X¹⁷+4X¹⁹；

(3), definition removes formula, with 1,2,4,8,16,32,3,6,12,24 for polynomial zero point, makes g (X)=25X⁰+28X¹ +42X²+33X³+63X⁴+46X⁵+26X⁶+45X⁷+7X⁸+46X⁹+X¹⁰It is 0 in above-mentioned zero point；

(4) identity information multinomial, is obtained into check polynomial, r (X)=34X with removing formula modulo operation⁰+48X¹+13X² +58X³+45X⁴+7X⁵+6X⁶+22X⁷+40X⁸+18X⁹, check polynomial is subtracted from identity information multinomial, obtains outer packing volume Code polynomial is C (X)=34X⁰+48X¹+13X²+58X³+45X⁴+7X⁵+6X⁶+22X⁷+40X⁸+18X⁹+44X¹⁰+54X¹¹+11X¹²+ 49X¹³+9X¹⁴+7X¹⁵+8X¹⁶+7X¹⁷+4X¹⁹；

(5) it is, outer by order inverse mapping of each term coefficient of outer packing coding polynomial according to polynomial order from low to high Packager code character string, it is YmDwj76MeIisBn978704 that can obtain outer packing coded string according to Fig. 1 mapping relations, outside Check character substring and identity information character substring form be as shown in Figure 5 in packager code character string.

2nd, specific decoding step (as shown in Figure 6)

(6), check end and read outer packing character string, in character string because damaged and the character of None- identified is accorded with underscore Number _ represent, in embodiment one, the information of outer packing coded string residual is YmDwj76__________704；

(7) it is 10 that, decoder module, which detects to obtain destroyed character quantity, is not above entangling for decoder module and deletes model Enclose, destroyed position of the character in outer packager code character string is followed successively by the 7th, 8,9,10,11,12,13,14,15,16 word Accord with (counting since 0), be then mapped as deleting symbol multinomial C'(X by remaining outer packing coded string)=34X⁰+48X¹+ 13X²+58X³+45X⁴+7X⁵+6X⁶+7X¹⁷+4X¹⁹, the character coefficient of correspondence destroyed is set to 0；

(8) the deletion code polynomial E (X) that destroyed character is formed, is solved, it is assumed that the following E of the polynomial that deletes code (X) =e₀X⁷+e₁X⁸+e₂X⁹+e₃X¹⁰+e₄X¹¹+e₅X¹²+e₆X¹³+e₇X¹⁴+e₈X¹⁵+e₉X¹⁶, wherein e₀,e₁,e₂…e₉Represent what is be destroyed Character coefficient of correspondence, E (X) and C'(X) will be brought into successively except the zero point of formula, obtain system of linear equations Ax=b, wherein matrix A j Row i row are defined asα^jRepresent j-th of zero point, β_iRepresent the position of i-th of destruction character, j-th yuan of right-hand vector Element is defined as b_j=-C'(α^j), x solves system of linear equations to delete code polynomial unknowm coefficient vector, obtains deleting code multinomial Formula E (X), it is in this example E (X)=22X⁷+40X⁸+18X⁹+44X¹⁰+54X¹¹+11X¹²+49X¹³+9X¹⁴+7X¹⁵+8X¹⁶；

(9) E (X) is superimposed upon C'(X) on, you can obtain outer packing coding polynomial C (X), C (X)=34X⁰+48X¹+ 13X²+58X³+45X⁴+7X⁵+6X⁶+22X⁷+40X⁸+18X⁹+44X¹⁰+54X¹¹+11X¹²+49X¹³+9X¹⁴+7X¹⁵+8X¹⁶+7X¹⁷+ 4X¹⁹, each term coefficient of multinomial is deployed to obtain finite state integer sequence, outer packing coded string is obtained after inverse mapping YmDwj76MeIisBn978704, extraction can check merchandise news after obtaining identity information character string isBn978704.

The technological thought of above example only to illustrate the invention, it is impossible to protection scope of the present invention is limited with this, it is every According to technological thought proposed by the present invention, any change done on the basis of technical scheme, the scope of the present invention is each fallen within Within；The technology that the present invention is not directed to can be realized by prior art.

Claims

1. a kind of delete string encoding and coding/decoding method based on entangling for finite state conversion, it is characterised in that：The step of this method It is rapid as follows：

(2) each character in identity information character string, is mapped as a nonnegative integer for showing its state, and will be non-negative whole Number obtains the finite state sequence a that a length is 10 according to the sequencing of character₀,a₁…a₉, it is identity to make each nonnegative integer Each term coefficient of message polynomial, then identity information multinomial be

(4), identity information multinomial and formula modulo operation will be removed, obtains check polynomial r (X), r (X)=m (X) modg (X), Check polynomial is subtracted from identity information multinomial, obtains outer packing coding polynomial C (X)=m (X)-r (X)；

(5), each term coefficient of outer packing multinomial is encoded according to the order inverse mapping of polynomial order from low to high for outer packing Character string；

(6), check end and read outer packing coded string, reading program has three according to the destruction situation of outer packing coded string Kind countermeasure：1. without breakage, directly extract identity information character string and obtain the progress commodity examination of its identity information；2. outer packing Coded string breakage calls core codec module Exact recovery to go out original outer packing code character within the scope of it can entangle and delete String, then identity information character string is extracted, carry out commodity examination；3. the destruction character quantity of outer packing coded string is beyond decoding The accurate decoding ranges of module, decoder return to error message code, terminate decoding effort；

(7), such as outer packing coded string, which exists, destroys character and destroys quantity within the scope of accurate decoding, decoder module inspection Destroyed character quantity N is measured, is destroyed position β of the character in outer packager code character string₀,β₁…β_N-1, and will be surplus Remaining character string maps are to delete symbol multinomial C'(X), the character coefficient of correspondence destroyed is set to 0；

(8) the deletion code polynomial E (X) that destroyed character is formed, is solved, if destroyed character coefficient of correspondence is followed successively by e₀, e₁,e₂…e_N-1, there is deletion code polynomialMay be such that C (X)=C'(X)+E (X), C is obtained by step (4) (X) root is α⁰,α¹…α⁹, then C (α^m)=C'(α^m)+E(α^m)=0, m=0,1 ... 9, then E (α^m)=- C'(α^m), by α⁰, α¹…α^N-1Bring E (X) and C'(X) into successively, obtain system of linear equations Ax=b, the j rows i row of wherein matrix A are defined asJ-th of element definition of right-hand vector is b_j=-C'(α^j), x solves line to delete code polynomial unknowm coefficient vector Property equation group, obtain delete code polynomial E (X)；

(9) E (X) is superimposed upon C'(X) on, you can outer packing coding polynomial C (X) is obtained, outer packing coding polynomial is put Enter each term coefficient to deploy to obtain finite state integer sequence, outer packing coded string is obtained after inverse mapping, extraction obtains identity Merchandise news can be checked after information character string.

2. according to claim 1 delete string encoding and coding/decoding method based on entangling for finite state conversion, its feature It is：Described step (1)-(5) are cataloged procedure, and the process can be to 1.15 × 10¹⁸Individual identity information string encoding life Into outer packing coded string.

3. according to claim 1 or 2 delete string encoding and coding/decoding method based on entangling for finite state conversion, it is special Sign is：Outer packing coded string in the step (5) is made up of 20 characters, and wherein top ten character is to check character String, rear ten characters are identity information character string.

4. according to claim 1 or 2 delete string encoding and coding/decoding method based on entangling for finite state conversion, it is special Sign is：Identity information character string in the step (1) by 10 Arabic numerals 0-9,26 English capitalization A-Z, Wantonly ten compositions in 26 English lower case a-z and 2 additional character-#.

5. according to claim 4 delete string encoding and coding/decoding method based on entangling for finite state conversion, its feature It is：The alphabet in outer packing coded string in the step (5) is by 10 Arabic numerals 0-9,26 English 20 compositions in capitalization A-Z, 26 English lower case a-z and 2 additional character-#.

6. according to claim 1 delete string encoding and coding/decoding method based on entangling for finite state conversion, its feature It is：Identity information character string in the step (1) is used to check institute by the identity information character string in traceability system The production of the commodity of reference, circulation information.

7. according to claim 1 delete string encoding and coding/decoding method based on entangling for finite state conversion, its feature It is：The formula exponent number that removes defined in the step (3) is 10, and the multinomial is using ten different finite state integers as multinomial Simple zero.

8. according to claim 1 delete string encoding and coding/decoding method based on entangling for finite state conversion, its feature It is：Character string breakage in the step (6) can entangle delete within the scope of refer to impaired outer packing coded string Character quantity be no more than 10.

9. deleting string encoding and coding/decoding method based on entangling for finite state conversion according to claim 1 or 8, it is special Sign is：Described step (6)-(9) are decoding process, and the decoding process can be to the word of impaired outer packing coded string Accord with outer packing coded string of the quantity no more than 10 and carry out accurate recovery, decoding.

10. string encoding and coding/decoding method are deleted based on entangling for finite state conversion according to claim 1 or 8, its It is characterised by：The decoding process of step (6)-(9) only with outer packing coded string to be damaged character quantity relevant.