CN106788454B - Construction method of local unequal codes - Google Patents

Construction method of local unequal codes Download PDF

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CN106788454B
CN106788454B CN201611071571.XA CN201611071571A CN106788454B CN 106788454 B CN106788454 B CN 106788454B CN 201611071571 A CN201611071571 A CN 201611071571A CN 106788454 B CN106788454 B CN 106788454B
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CN106788454A (en
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朱丹
周晓波
朱洁
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Shaanxi Shangpin Information Technology Co ltd
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    • H—ELECTRICITY
    • H03—ELECTRONIC CIRCUITRY
    • H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
    • H03M13/00—Coding, decoding or code conversion, for error detection or error correction; Coding theory basic assumptions; Coding bounds; Error probability evaluation methods; Channel models; Simulation or testing of codes
    • H03M13/03—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words
    • H03M13/05—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words using block codes, i.e. a predetermined number of check bits joined to a predetermined number of information bits
    • H03M13/09—Error detection only, e.g. using cyclic redundancy check [CRC] codes or single parity bit

Abstract

The invention discloses a construction method of local unequal codes, which has the following implementation scheme: under the condition of unequal locality of information symbols, firstly, a parity check bit matrix of maximum distance separable codes MDS is separated, and then any subset obtained by separation is further separated according to a certain rule to obtain unequal locality of information symbols. The case of unequal locality of all symbols is first to
Figure DDA0001165284900000011
Vector on domain
Figure DDA0001165284900000012
Gabidulin coding is carried out, then two rounds of division are carried out on the coding result, and the maximum distance divisible code MDS coding is carried out on the result of the second division to obtain the full symbol locality unequal code. The invention effectively solves the problem that the minimum distance of the existing local repair code can not reach the upper bound, correspondingly reduces the repair locality and the I/O operation in the repair process, and keeps a lower repair bandwidth in the repair process.

Description

Construction method of local unequal codes
Technical Field
The invention belongs to the field of data coding and storage, and particularly relates to a construction method of local unequal codes.
Background
In recent years, with the rapid development of portable internet devices and mobile internet, the demand for storage has been increasing. Distributed storage is emerging as the primary solution to mass storage, which stores data in a distributed manner on multiple independent devices. In order to save cost, the nodes usually adopt common switching equipment and cheap servers, so that the nodes in the storage system are very easy to fail under the conditions of node replacement, hardware failure and software upgrading, thereby causing the loss of data in the storage nodes. In order to ensure data reliability, the system frequently performs node repair, and therefore how to effectively perform node repair becomes an urgent problem to be solved. Now, a series of schemes adopted by the repair node, such as a copy policy, a regeneration code, an erasure code, etc., may cause a waste of storage space, occupy a large amount of repair bandwidth, and require a large repair locality (i.e., a large number of other nodes that need to be connected in the data node repair process).
The local repair code LRCs means that all information symbols of one code are divided into a plurality of groups, and each group generates one parity bit. When an information symbol in a certain packet is lost or damaged, the lost or damaged data can be recovered by only using other information symbols in the group and the parity check bits of the group, and the recovery is not required by all the coded symbols. Therefore, the local repair codes LRCs can reduce the repair locality, so that the I/O operation in the repair process can be reduced, and a lower repair bandwidth can be kept.
For a local repair code LRCs, the larger its minimum distance d, the fewer the number of nodes that need to be connected in the data node repair process, and the smaller the repair bandwidth. In the existing method for constructing the local repair code, the minimum distance of the local repair code can rarely reach an upper limit.
Disclosure of Invention
The invention provides a new construction method of local unequal codes, aiming at the defect that the existing construction method of local repair codes LRCs has larger repair locality. And constructing a new local unequal code based on two conditions that the local repair code information symbols are unequal in locality and the full symbols are unequal in locality.
The repair locality of the locally unequal codes is minimized, and the technical scheme of the invention comprises the following steps:
a construction method of local unequal codes comprises two preferred embodiments:
first preferred embodiment: the method is characterized in that the locality of information symbols is unequal, information symbol locality unequal codes are constructed, a parity check bit matrix of the MDS code with the maximum distance divisible is separated, any subset obtained by separation is further separated, an information symbol locality unequal code generating matrix with the code symbol length n is obtained, and a code word generated by the matrix is an information symbol locality unequal code with the minimum distance d reaching the upper bound;
second preferred embodiment: the method is characterized in that the full symbol locality is unequal, the full symbol locality comprises an information symbol and a parity check symbol, Gabidulin codes are adopted for coding, coding results are coded again by using maximum distance separable MDS codes, a (n, k, d) full symbol locality unequal code reaching the upper bound of the minimum distance is obtained, the length of a code word is n, and the code word generated by the matrix is the information symbol locality unequal code reaching the upper bound of the minimum distance d.
Further according to the method for constructing the locally unequal codes, the method comprises the following steps:
Fqrepresenting a q-ary domain;
Figure BDA0001165284880000021
wherein F represents a domain, qmRepresenting an m-th order expansion of q-elements, wherein the highest order of the polynomial in the domain is k-1;
a code of length n contains k information symbols, where a certain symbol i may pass through the other r symbols in the codeiThe symbols are recovered, then the locality of i is riIf the locality of each information symbol in k information symbols included in one code is maximum r, the locality of the code is r;
system code
Figure BDA0001165284880000022
n denotes the code length, k denotes the information symbol length, d denotes the minimum distance, if the information symbols can be divided into disjoint subsets, the information symbols of different subsets have different locality, i.e. the code is an information symbol locality unequal code.
The system code
Figure BDA0001165284880000023
The information symbol locality profile is
Figure BDA0001165284880000024
kjThe number of information symbols with locality j (j is more than or equal to 1 and less than or equal to r);
the full symbol locality inequality code is a further extension of the information symbol locality inequality code, and the coded symbols, including the information symbols and the parity check symbols, can be divided into disjoint subsets, the coded symbols of different subsets have different locality, and represent the codes of which the parity check symbols also have locality constraint;
the system code
Figure BDA0001165284880000025
If the parity check symbols also have locality constraints, a full symbol locality profile is defined, similar to the information symbol locality profile, riRepresenting the locality of the ith symbol in the code, i is more than or equal to 1 and less than or equal to n, and r isa＝max(r1，r2，…，rn) Then its full-symbol locality profile is represented as
Figure BDA0001165284880000026
njIs localized as j (1 ≦ j ≦ ra) The number of information symbols of (a);
if XnIs an n-dimensional vector space over a finite field GF (q), q being a prime number or a prime power, XnThe rank of the element vector X in GF (q) is R (X), XnX, y of the two elements x, y of (a) is a rank distance dR(X, Y) is defined as dR(x, y) ═ R (x-y); the minimum value of the rank distances of all two different codewords of the code c is the minimum rank distance of the code c and is denoted as dR(c) (ii) a A linear code with a code length of N, an information symbol number of K, and a minimum rank distance of D is called a rank distance (N, K, D) code;
the above-mentioned
Figure BDA00011652848800000210
The above Gabidulin code is denoted as (N, K, N-K +1) code, where N is the number of coding symbols, K is the number of information symbols, and N-K +1 is the minimum distance of code words.
Figure BDA0001165284880000027
Is one of the code words, defined as
Figure BDA0001165284880000028
f (x) is within the order of m × qOne coefficient of (a) is a linear polynomial of the information symbol, i.e. the number of elements in the finite field is m x q, g1，…，gNIs that
Figure BDA0001165284880000029
A particular point on;
the Gabidulin code belongs to a rank distance code;
if the code length of the (N, N-Y, Y +1) code is N, the redundancy is Y, and the linear code with the minimum distance of Y +1 is the maximum distance separable MDS code; if N-Y information sign bits are expanded into N information sign bits after being coded, when any sign bit of the N-Y information sign bits is lost or damaged, K sign bits in the existing N-1 sign bits can be used for recovering the lost or damaged information sign bits;
if Gabidulin codes are adopted for coding, then maximum distance separable MDS codes are adopted for coding, and the finally obtained coding result can reach the upper bound of Singleton, namely the upper bound of the minimum distance of the code words;
the Singleton upper bound is a measure of the code word, and is an upper bound of the number of the code words when the length of the code word and the minimum distance are given; a. theq(n, d) is the maximum value that the number of q element code words can reach, namely Aq(n，d)≤qn-d+1Wherein q represents that the codeword is a q-ary code; if a code word reaches the Singleton upper bound, the minimum distance of the code word can reach the maximum value; if it is
Figure BDA0001165284880000031
(N, K, D) rank distance code over field
Figure BDA0001165284880000032
The number of coding symbols is N, the number of information symbols is K, the minimum rank distance is D, and the Singleton upper bound can be converted into a representation related to the minimum distance of the code words, namely:
Figure BDA0001165284880000033
further according to the construction method of the locally unequal codes, when the information symbols in the first preferred embodiment are locally unequal;
code
Figure BDA0001165284880000034
Wherein k + d-1 represents the coding length, d-1 represents the parity check bit length, d represents the minimum distance, if the parity check bit length is less than the minimum distance by one bit, the construction condition of the maximum distance separable MDS code is met, the generating matrix of the maximum distance separable MDS is constructed by a unit matrix and a parity check bit matrix, wherein the column number of the unit matrix is the same as the information symbol length of the code;
code
Figure BDA0001165284880000035
Can be expressed as
Figure BDA0001165284880000036
Wherein
Figure BDA0001165284880000037
Is the jth column vector of the identity matrix of k x k,
Figure BDA0001165284880000038
is the j-th column vector in the parity check bit matrix of k (d-1), i.e.:
Figure BDA0001165284880000039
of the locality pairs G' of each information symbol
Figure BDA00011652848800000310
Dividing the subsets, and keeping jp as p (l is less than or equal to p and less than or equal to m, and m is less than or equal to r) in the j column of the parity check bit matrix, the locality of the coordinate point is represented by p
Figure BDA00011652848800000311
All coordinates with locality p in the same subset spThen, then
Figure BDA00011652848800000312
Figure BDA00011652848800000313
Is divided into m disjoint subsets s1，…，sm，|spI represents the set spThe number of the elements in the Chinese character,
Figure BDA00011652848800000314
denotes jpThe number of (2);
will spArbitrarily divided into
Figure BDA00011652848800000315
Disjoint subsets, each subset not exceeding the locality p, i.e.
Figure BDA00011652848800000316
Figure BDA00011652848800000317
Is a k-dimensional vector, the set S is contained in this k-dimensional vector,
Figure BDA00011652848800000318
the elements in the representation set S are taken from
Figure BDA00011652848800000319
Any | S | row of (1);
the resulting codeword generator matrix is G:
Figure BDA00011652848800000320
the decomposition vector
Figure BDA0001165284880000041
Taking this as an example only, but not limiting to this example;
if the original information code word is
Figure BDA0001165284880000042
Obtain a code word of
Figure BDA0001165284880000043
The obtained codeword length n:
Figure BDA0001165284880000044
upper bound of minimum distance d:
Figure BDA0001165284880000045
then the information symbol locality inequality codes with the minimum distance d reaching the upper bound are constructed.
Further according to the construction method of the locally unequal code, when the all symbols in the second preferred embodiment are locally unequal, the method includes an information symbol and a parity check symbol;
vector quantity
Figure BDA0001165284880000046
Belong to
Figure BDA0001165284880000047
A field, having a length of k,
Figure BDA0001165284880000048
the full-symbol locality profile of
Figure BDA0001165284880000049
I.e. the number of symbols with locality j is njJ is more than or equal to 1 and less than or equal to ra;
Figure BDA00011652848800000410
finding n corresponding to each locality jjIs of the formula
Figure BDA00011652848800000411
Further solving N to obtain the code word length N of the Gabidulin code;
according to the code word length N, the information symbol length k and the minimum distance N-k +1 pair
Figure BDA00011652848800000412
Gabidulin coding is carried out to obtain code words
Figure BDA00011652848800000413
Code word
Figure BDA00011652848800000414
The symbols in (1) are divided into r according to the locality of each symbolaA disjoint group
Figure BDA00011652848800000415
I.e. the locality of the elements in each group is j, each group
Figure BDA00011652848800000416
The number of the middle symbols is Nj；
If N is presentjWhen the value is 0, then group
Figure BDA00011652848800000417
Such groups are not further divided;
if N is presentj> 0, i.e. group
Figure BDA00011652848800000418
Then further N is addedjCorresponding group
Figure BDA00011652848800000419
The symbols in (1) are arbitrarily divided into Nj/j disjoint locality groups, the number of symbols in each group is j, then
Figure BDA00011652848800000420
For each locality grouping of j symbols
Figure BDA00011652848800000421
Using FqThe (j +1, j,2) maximum distance separable MDS codes are coded again, so that the number of symbols of each group is changed from j to j + 1;
obtaining (n, k, d) full symbol locality inequality codes reaching a minimum distance upper bound, a codeword length n:
Figure BDA00011652848800000422
upper bound of minimum distance d:
Figure BDA0001165284880000051
a full symbol locality inequality code is constructed with a minimum distance d up to the upper bound.
Compared with the prior art, the invention has the following advantages:
(1) the invention adopts the construction method of local unequal codes, so that the repair locality of the code words reaches a minimum value, thereby reducing I \ O operation and reducing repair bandwidth.
(2) Because the locality inequality codes divide the unequal locality, a locality outline is formed, and the basis for dividing subsets of code words for multiple times is given, so that the minimum distance d reaches the upper bound.
Drawings
Fig. 1 shows two preferred modes of locally unequal codes.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more clear, some definitions and parameters related to the method of the present invention are first explained as follows:
Fq: representing a q-ary domain.
Figure BDA0001165284880000052
F denotes a field, qmRepresenting an m-th order extension of q-elements, where the highest order of the polynomial in the field is k-1.
The locality is as follows: setting a code of length n to contain k information symbolsWhere a certain symbol i may pass through the others r in the codeiThe symbols are recovered, then the locality of i is ri. If the locality of each of k information symbols included in a code is at most r, the locality of the code is said to be r.
Information symbol locality unequal code: for a system code
Figure BDA0001165284880000053
n denotes a code length, k denotes an information symbol length, and d denotes a minimum distance. If the information symbols can be divided into disjoint subsets, the information symbols of different subsets have different locality, and the code is called an information symbol locality inequality code.
Information symbol locality profile: for a system code
Figure BDA0001165284880000054
Its information symbol locality profile is expressed as
Figure BDA0001165284880000055
kjThe number of information symbols having locality j (1. ltoreq. j. ltoreq.r) is represented.
Full symbol locality unequal code: a full symbol locality unequal code is a further extension of an information symbol locality unequal code to mean a code where the parity check symbols also have locality constraints. For a systematic code, if its code symbols (including information symbols and parity check symbols) can be divided into disjoint subsets, the code symbols of different subsets have different locality, such a code is called a full symbol locality unequal code.
Full-symbol locality profile: when the parity symbols also have locality constraints, a full symbol locality outline may be defined, similar to the information symbol locality outline. Firstly, for a system code
Figure BDA0001165284880000056
riRepresenting the locality of the ith symbol in the code, i is more than or equal to 1 and less than or equal to n, and then r is madea＝max(r1，r2，…，rn) Then its full-symbol locality profile is represented as
Figure BDA0001165284880000057
njIs localized as j (1 ≦ j ≦ ra) The number of information symbols of (2).
Rank distance: let XnRepresents an n-dimensional vector space over the finite field gf (q), where q is a prime number or a prime power. XnThe rank of the element vector x over gf (q) is r (x). XnX, y of the two elements x, y of (a) is a rank distance dR(X, Y) is defined as dR(x, y) ═ R (x-y). The minimum value of the rank distances of all two different codewords of the code c is the minimum rank distance of the code c and is denoted as dR(c) In that respect A linear code with a code length of N, an information symbol number of K, and a minimum rank distance of D is called a rank distance (N, K, D) code.
Gabidulin code:
Figure BDA0001165284880000061
the above Gabidulin code is denoted as (N, K, N-K +1) code, where N is the number of coding symbols, K is the number of information symbols, and N-K +1 is the minimum distance of code words.
Figure BDA0001165284880000062
Is one of the code words, defined as
Figure BDA0001165284880000063
(x) is a linear polynomial with the information symbol as a coefficient in a finite field of order m x q (i.e. the number of elements in the finite field is m x q), g1，…，gNIs that
Figure BDA0001165284880000064
A particular point on. The Gabidulin code belongs to rank distance codes.
MDS code: one such (N, N-Y, Y +1) code is called a maximum distance separable MDS code, which is a linear code with a code length N, redundancy Y, and minimum distance Y + 1. It has the following properties: if N-Y information sign bits are expanded into N information sign bits after being coded, when any sign bit in the N-Y information sign bits is lost or damaged, the lost or damaged information sign bit can be recovered by only using K sign bits in the existing N-1 sign bits.
If Gabidulin code is adopted for coding, then maximum distance separable MDS code is adopted for coding, the finally obtained coding result can reach the upper bound of Singleton, and the upper bound of the minimum distance of the code words is also reached.
Singleton upper bound: the Singleton upper bound is a measure of the number of codewords, which is an upper bound on the number of codewords given the codeword length and minimum distance. The Singleton upper bound gives the relation between the length of the codeword and the minimum distance, Aq(n, d) represents the maximum possible number of q-element code words, namely Aq(n，d)≤qn-d+1Where q denotes that the codeword is a q-ary code. When a code reaches the Singleton upper bound, the minimum distance of this codeword may reach a maximum. One is
Figure BDA0001165284880000065
(N, K, D) rank distance code over field
Figure BDA0001165284880000066
The number of coding symbols is N, the number of information symbols is K, and the minimum rank distance is D. Such codes, the Singleton upper bound of which can be translated into a representation related to the minimum distance of the codeword, i.e. a code with a minimum distance between the codewords
Figure BDA0001165284880000067
In order to make the objects, technical solutions and advantages of the present invention more apparent, a method for constructing a locally unequal code according to the present invention is described in detail below with reference to the accompanying drawings, as shown in fig. 1, the method according to the present invention preferably includes a first preferred embodiment and a second preferred embodiment.
First preferred embodiment:
the first preferred embodiment is applicable to cases where the locality of information symbols is unequal, and in order to construct information symbol locality unequal codes, first, the first preferred embodiment is to construct information symbol locality unequal codesFirstly, the parity check bit matrix of the maximum distance separable MDS code is separated, then any subset obtained by separation is further separated, and the length of the code symbol is obtained
Figure BDA0001165284880000068
The information symbol locality inequality code of (2) generates a matrix, and a codeword generated by the matrix is an information symbol locality inequality code of which the minimum distance d reaches an upper bound.
Given a code
Figure BDA0001165284880000069
Wherein k + d-1 represents the coding length, d-1 represents the parity check bit length, d represents the minimum distance, when the parity check bit length is less than the minimum distance by one bit, the construction condition of the maximum distance separable MDS code is satisfied, the generation matrix of the maximum distance separable MDS is constructed by a unit matrix and a parity check bit matrix, wherein the column number of the unit matrix is the same as the information symbol length of the code, so that the code has the advantages of simple structure, low cost, high reliability, and high reliability
Figure BDA00011652848800000610
Can be expressed as
Figure BDA00011652848800000611
Wherein
Figure BDA00011652848800000612
Is the jth column vector of the identity matrix of k x k,
Figure BDA00011652848800000613
is the j-th column vector in the parity check bit matrix of k x (d-1). Namely, it is
Figure BDA0001165284880000071
According to locality of each information symbol in G
Figure BDA0001165284880000072
Dividing the subsets into subsets, note jpThe locality of the representative coordinate point in the jth column of the parity check bit matrix is p (1 ≦ p ≦ m, m ≦ r). Specifically, will
Figure BDA0001165284880000073
All coordinates with locality p in the same subset spThen, then
Figure BDA0001165284880000074
|spI represents the set spThe number of the elements in the Chinese character,
Figure BDA0001165284880000075
denotes jpThe number of (2).
Figure BDA0001165284880000076
Is divided into m disjoint subsets s1，…，sm。
Will spArbitrarily divided into
Figure BDA0001165284880000077
Disjoint subsets, each subset not exceeding the locality p, i.e.
Figure BDA0001165284880000078
Here, a vector in k-dimension is defined
Figure BDA0001165284880000079
The set S is contained in this k-dimensional vector,
Figure BDA00011652848800000710
the elements in the representation set S are taken from
Figure BDA00011652848800000711
Arbitrary | S | row of (1). By decomposing the vector
Figure BDA00011652848800000712
For example, and not by way of limitation, the finally obtained codeword generator matrix is G:
Figure BDA00011652848800000713
if the original information code word is
Figure BDA00011652848800000714
Then the coded codeword is obtained as
Figure BDA00011652848800000715
The obtained codeword length n:
Figure BDA00011652848800000716
the upper bound of the minimum distance d is:
Figure BDA00011652848800000717
then the information symbol locality inequality codes with the minimum distance d reaching the upper bound are constructed.
Second preferred embodiment:
the second preferred embodiment is applicable to cases where the full symbol locality is unequal, including information symbols and parity check symbols, and the coding is performed using the Gabidulin code first, and then the coding result is encoded again using the maximum distance separable MDS code.
Select one to
Figure BDA00011652848800000718
Vector of domain
Figure BDA00011652848800000719
The length of the first electrode is k,
Figure BDA00011652848800000720
the full-symbol locality profile of
Figure BDA00011652848800000721
I.e. the number of symbols with locality j is nj1 is more than or equal to j is less than or equal to ra. Order to
Figure BDA00011652848800000722
Thereby obtaining n corresponding to each locality jjThen run by
Figure BDA00011652848800000723
And further solving N to obtain the code word length N of the Gabidulin code. Then according to the code word length N, the information symbol length k and the minimum distance N-k +1 pair
Figure BDA0001165284880000081
Gabidulin coding is carried out to obtain code words
Figure BDA0001165284880000082
Coding the word
Figure BDA0001165284880000083
The symbols in (1) are divided into r according to the locality of each symbolaA disjoint group
Figure BDA0001165284880000084
I.e. the locality of the elements in each group is j, each group
Figure BDA0001165284880000085
The number of the middle symbols is Nj. If N is presentjWhen the value is 0, then group
Figure BDA0001165284880000086
Such groups are not further divided; if N is presentj> 0, i.e. group
Figure BDA0001165284880000087
Then further N is addedjCorresponding group
Figure BDA0001165284880000088
The symbols in (1) are arbitrarily divided into Nj/j disjoint locality groups, the number of symbols in each group is j, then
Figure BDA0001165284880000089
For each locality grouping of j symbols
Figure BDA00011652848800000810
By using EqThe above (j +1, j,2) maximum distance separable MDS code is encoded again, so that the number of symbols per packet is changed from j to j + 1. This results in (n, k, d) full symbol locality unequal code reaching a minimum distance upper bound, codeword length:
Figure BDA00011652848800000811
upper bound of minimum distance d:
Figure BDA00011652848800000812
a full symbol locality inequality code is constructed with a minimum distance d up to the upper bound.
The above examples are merely illustrative for clearly illustrating the present invention and are not intended to limit the embodiments. Other variations and modifications will be apparent to persons skilled in the art in light of the above description. It is not necessary or necessary to exhaustively enumerate all embodiments herein, and obvious variations or modifications can be made without departing from the scope of the invention.

Claims (1)

1. A method for constructing locally unequal codes, comprising:
the method comprises the following steps: when the locality of the information symbols is unequal, constructing an unequal locality code of the information symbols, separating a parity check bit matrix of the MDS code with the maximum distance, further separating any subset obtained by separation, obtaining an unequal locality code generating matrix of the information symbols with the code symbol length n, wherein a code word generated by the matrix is the unequal locality code of the information symbols with the minimum distance d reaching the upper bound:
code
Figure FDA0002370438590000011
When the length of the parity check bit is only one bit less than the minimum distance, the construction condition of the maximum distance separable MDS code is met, a generating matrix of the maximum distance separable MDS code is constructed by a unit matrix and a parity check bit matrix, wherein the column number of the unit matrix is the same as the length of the information symbols of the maximum distance separable MDS code;
code
Figure FDA0002370438590000012
Can be expressed as
Figure FDA0002370438590000013
Wherein
Figure FDA0002370438590000014
Is the jth column vector of the identity matrix of k x k,
Figure FDA0002370438590000015
is the j-th column vector in the parity check bit matrix of k (d-1), i.e.:
Figure FDA0002370438590000016
of the locality pairs G' of each information symbol
Figure FDA0002370438590000017
Dividing the subsets into subsets, note jpThe locality of the representative coordinate point in the jth column of the parity check bit matrix is p (1 ≦ p ≦ m, m ≦ r), and the representative coordinate point is represented by p
Figure FDA0002370438590000018
All coordinates with locality p in the same subset spThen, then
Figure FDA0002370438590000019
Figure FDA00023704385900000110
Is divided into m disjoint subsets s1，…，sm，|spI represents the set spThe number of the elements in the Chinese character,
Figure FDA00023704385900000111
denotes jpThe number of (2);
will spArbitrarily divided into
Figure FDA00023704385900000112
Disjoint subsets, the number of elements in each subset not exceeding the locality p, i.e.
Figure FDA00023704385900000113
Figure FDA00023704385900000114
Is a k-dimensional vector, the set S is contained in this k-dimensional vector,
Figure FDA00023704385900000115
the elements in the representation set S are taken from
Figure FDA00023704385900000116
Any | S | row of (1);
by decomposing the vector
Figure FDA00023704385900000117
For example, the finally obtained codeword generator matrix is G:
Figure FDA00023704385900000118
if the original information code word is
Figure FDA00023704385900000119
Obtain a code word of
Figure FDA00023704385900000120
The obtained codeword length n:
Figure FDA00023704385900000121
upper bound of minimum distance d:
Figure FDA0002370438590000021
constructing the information symbol locality inequality code with the minimum distance d reaching the upper bound;
step two: when the full symbols are not equal in locality, the full symbols comprise information symbols and parity check symbols, Gabidulin codes are adopted for coding, coding results are coded again by using maximum distance separable MDS codes, and (n, k, d) full symbol unequal codes reaching the upper bound of the minimum distance, the length n of the code words and the length k of the information symbols are obtained, and the code words generated by the matrix are the information symbol unequal codes reaching the upper bound of the minimum distance d:
vector quantity
Figure FDA0002370438590000022
Belong to
Figure FDA0002370438590000023
A field, having a length of k,
Figure FDA0002370438590000024
the full-symbol locality profile of
Figure FDA0002370438590000025
I.e. symbols with locality jThe number is nj1 is more than or equal to j is less than or equal to ra；
Figure FDA0002370438590000026
Determining what each locality j corresponds to jNIs of the formula
Figure FDA0002370438590000027
Further solving N to obtain the code word length N of the Gabidulin code; according to the code word length N, the information symbol length k and the minimum distance N-k +1 pair
Figure FDA0002370438590000028
Gabidulin coding is carried out to obtain code words
Figure FDA0002370438590000029
Code word
Figure FDA00023704385900000210
The symbols in (1) are divided into r according to the locality of each symbolaA disjoint group
Figure FDA00023704385900000211
I.e. the locality of the elements in each group is j, each group
Figure FDA00023704385900000212
The number of the middle symbols is Nj；
If N is presentjWhen the value is 0, then group
Figure FDA00023704385900000213
Such groups are not further divided;
if N is presentj> 0, i.e. group
Figure FDA00023704385900000214
Then further N is addedjCorresponding group
Figure FDA00023704385900000215
Symbol in (1) is arbitrarily divided into NjA/j number of different phases
Figure FDA00023704385900000216
Coding again to change the number of symbols of each group from j to j + 1;
obtaining (n, k, d) full symbol locality inequality codes reaching a minimum distance upper bound, a codeword length n:
Figure FDA00023704385900000217
upper bound of minimum distance d:
Figure FDA00023704385900000218
constructing a full-symbol partial inequality code with the minimum distance d reaching an upper bound;
wherein, F isqRepresenting a q-ary domain;
the above-mentioned
Figure FDA00023704385900000219
Wherein F represents a domain, qmRepresenting an m-th order expansion of q-elements, wherein the highest order of the polynomial in the domain is k-1;
a code of length n contains k information symbols, where a certain symbol i may pass through the other r symbols in the codeiThe symbols are recovered, then the locality of i is riIf the locality of each information symbol in k information symbols included in one code is maximum r, the locality of the code is r;
system code
Figure FDA0002370438590000031
n denotes the code length, k denotes the information symbol length, d denotes the minimum distance, if the information symbols can be divided into disjoint subsets, the information symbols of different subsets have different locality, i.e. the information symbolsInformation symbol locality unequal codes;
the system code
Figure FDA0002370438590000032
The information symbol locality profile is
Figure FDA0002370438590000033
kjThe number of information symbols with locality j (j is more than or equal to 1 and less than or equal to r);
the full symbol locality inequality code is a further extension of the information symbol locality inequality code, and the coded symbols, including the information symbols and the parity check symbols, can be divided into disjoint subsets, the coded symbols of different subsets have different locality, and represent the codes of which the parity check symbols also have locality constraint;
the system code
Figure FDA0002370438590000034
If the parity check symbols also have locality constraints, then a full-symbol locality profile is defined, riRepresenting the locality of the ith symbol in the code, i is more than or equal to 1 and less than or equal to n, and r isa＝max(r1，r2，…，rn) Then its full-symbol locality profile is represented as
Figure FDA0002370438590000035
njIs localized as j (1 ≦ j ≦ ra) The number of information symbols of (a);
if XnIs an n-dimensional vector space over a finite field GF (q), q being a prime number or a prime power, XnThe rank of the element vector X in GF (q) is R (X), XnX, y of the two elements x, y of (a) is a rank distance dR(X, Y) is defined as dR(x, y) ═ R (x-y); the minimum value of the rank distances of all two different codewords of the code c is the minimum rank distance of the code c and is denoted as dR(c) (ii) a A linear code with a code length of N, an information symbol number of K, and a minimum rank distance of D is called a rank distance (N, K, D) code;
the above-mentioned
Figure FDA0002370438590000036
The Gabidulin code is marked as an (N, K, N-K +1) code, wherein N is the number of coding symbols, K is the number of information symbols, and N-K +1 is the minimum distance of code words;
Figure FDA0002370438590000037
is one of the code words, defined as
Figure FDA0002370438590000038
(x) is a linear polynomial with information symbols as a coefficient in a finite field of order m x q, i.e. the number of elements in the finite field is m x q, g1，…，gNIs that
Figure FDA0002370438590000039
A particular point on;
the Gabidulin code belongs to a rank distance code;
if the linear code satisfies that the code length is M, the redundancy is Y and the minimum distance is Y +1, the (M, M-Y, Y +1) code is a maximum distance separable MDS code; if M-Y information sign bits are expanded into M information sign bits after being coded, when any one of the M-Y information sign bits is lost or damaged, K sign bits in the existing M-1 sign bits can be used for recovering the lost or damaged information sign bits;
if Gabidulin codes are adopted for coding, then maximum distance separable MDS codes are adopted for coding, and the finally obtained coding result can reach the upper bound of Singleton, namely the upper bound of the minimum distance of the code words;
the Singleton upper bound is a measure of the codeword, i.e. an upper bound on the number of codewords when the codeword length and minimum distance are given; a. theq(n, d) is the maximum value that the number of q element code words can reach, namely Aq(n，d)≤qn-d+1Wherein q represents that the codeword is a q-ary code; if a code word reaches the Singleton upper bound, the minimum distance of the code word can reach the maximum value; if it is
Figure FDA00023704385900000310
(N, K, D) rank distance code over field
Figure FDA00023704385900000311
The number of coding symbols is N, the number of information symbols is K, and the minimum rank distance is D, then its Singleton upper bound can be converted into the expression of the minimum distance of the codeword, that is:
Figure FDA0002370438590000041
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