TWI688223B - Encoding and decoding method of algebraic geometric codes of Hermitian codes - Google Patents

Encoding and decoding method of algebraic geometric codes of Hermitian codes Download PDF

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TWI688223B
TWI688223B TW108104374A TW108104374A TWI688223B TW I688223 B TWI688223 B TW I688223B TW 108104374 A TW108104374 A TW 108104374A TW 108104374 A TW108104374 A TW 108104374A TW I688223 B TWI688223 B TW I688223B
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張耀祖
李崇道
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義守大學
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一種代數幾何碼的赫米特碼之編碼及解碼方法，傳送端根據參數資訊所產生的生成矩陣，其中參數資訊包括相關於不可約仿射平滑曲線的參數曲線方程式、有限體的元素，及預設度數。傳送端根據傳輸資料及生成矩陣產生包括傳輸資料的編碼資料，並經由傳輸通道傳送編碼資料至接收端。在接收到經傳輸通道之雜訊干擾的編碼資料後，接收端根據校驗矩陣獲得目標症狀值，並根據症狀值對錯誤樣式的查找表，獲得對應目標症狀值的錯誤樣式，並以錯誤樣式更正經傳輸通道之雜訊干擾的編碼資料的錯誤，以獲得傳輸資料。An encoding and decoding method for Hermitian codes of algebraic geometric codes. The generator generates a generating matrix based on parameter information. The parameter information includes parameter curve equations related to irreducible affine smooth curves, elements of finite bodies, and Set the degree. The transmitting end generates encoded data including the transmitted data according to the transmitted data and the generation matrix, and transmits the encoded data to the receiving end through the transmission channel. After receiving the encoded data of the noise interference through the transmission channel, the receiving end obtains the target symptom value according to the check matrix, and according to the symptom value lookup table for the error pattern, obtains the error pattern corresponding to the target symptom value, and uses the error pattern Correct the error of the coded data interfered by the noise of the transmission channel to obtain the transmission data.

Description

代數幾何碼的赫米特碼之編碼及解碼方法Encoding and decoding method of algebraic geometric codes of Hermitian codes

本發明是關於一種編碼及解碼方法，特別是指一種代數幾何碼的赫米特碼之編碼及解碼方法。The invention relates to an encoding and decoding method, in particular to an encoding and decoding method of algebraic geometric codes of Hermitian codes.

代數幾何碼(Algebraic geometry code, AG code)始於1980前後由前蘇聯數學家V.D.Goppa提出使用有限域上的代數曲線來構造碼，之後Tsfasman、Vlăduţ和Zink將Goppa的想法與代數幾何相結合，構造出一系列錯誤更正碼，其中，錯誤更正碼用在有雜訊通道中可靠地傳輸訊息，被廣泛應用於各領域。代數幾何碼在理論上已證明具有優越性能，不僅是漸近好碼(Shannon code)，也是唯一能超越Gilbert-Varshamov限的碼。Algebraic geometry code (AG code) began around 1980. Former Soviet mathematician VDGoppa proposed the use of algebraic curves on finite fields to construct codes. Later, Tsfasman, Vlăduţ, and Zink combined Goppa’s ideas with algebraic geometry. A series of error correction codes are constructed. Among them, the error correction codes are used to reliably transmit messages in noise channels and are widely used in various fields. Algebraic geometric codes have proved to have superior performance in theory, not only asymptotically good codes (Shannon codes), but also the only codes that can exceed the limit of Gilbert-Varshamov.

1990年起，各國學者著重於尋求代數幾何碼的高效率解碼算法，然而，現有的代數幾何碼的解碼算法是由原始里德-所羅門碼(Reed-Solomon code, RS code)的解碼算法推導而獲得，即代數幾何碼的解碼算法僅是從RS解碼算法加以改良，至今還沒有真正實用的解碼算法。Since 1990, scholars from various countries have focused on seeking efficient decoding algorithms for algebraic geometric codes. However, the existing decoding algorithms for algebraic geometric codes are derived from the decoding algorithm of the original Reed-Solomon code (RS code) Obtained, that is, the decoding algorithm of the algebraic geometric code is only improved from the RS decoding algorithm, so far there is no real practical decoding algorithm.

因此，本發明的目的，即在提供一種代數幾何碼之實用的編碼及解碼方法。Therefore, the object of the present invention is to provide a practical encoding and decoding method for algebraic geometric codes.

於是，本發明代數幾何碼的赫米特碼之編碼及解碼方法，由一系統所執行，該系統包括一傳送端及一接收端，該傳送端經由一傳輸通道傳輸一傳輸資料至該接收端，該接收端儲存有一症狀值對錯誤樣式的查找表，該查找表包括多個症狀值及多個相關於傳輸通道之雜訊且分別對應該等症狀值的錯誤樣式，該編碼及解碼方法包含一步驟(A)、一步驟(B)、一步驟(C)、一步驟(D)、一步驟(E)，及一步驟(F)。Therefore, the encoding and decoding method of the Hermitian code of the algebraic geometric code of the present invention is executed by a system including a transmitting end and a receiving end, the transmitting end transmits a transmission data to the receiving end through a transmission channel , The receiving end stores a lookup table of symptom value versus error pattern, the lookup table includes multiple symptom values and multiple noises related to the transmission channel and corresponding to the error pattern of the symptom values respectively, the encoding and decoding method includes One step (A), one step (B), one step (C), one step (D), one step (E), and one step (F).

該步驟(A)中，該傳送端根據所儲存的一參數資訊產生一生成矩陣，其中該參數資訊包括相關於位於三維座標系統的一不可約仿射平滑曲線的一參數曲線方程式、一有限體的多個元素，及一預設度數。In the step (A), the transmitting end generates a generating matrix according to a stored parameter information, wherein the parameter information includes a parameter curve equation and a finite body related to an irreducible affine smooth curve located in the three-dimensional coordinate system Multiple elements of, and a preset degree.

該步驟(B)中，該傳送端根據該傳輸資料及該生成矩陣產生包括該傳輸資料的一編碼資料。In the step (B), the transmitting end generates a coded data including the transmission data according to the transmission data and the generation matrix.

該步驟(C)中，該傳送端經由該傳輸通道傳送該編碼資料至該接收端。In step (C), the transmitting end transmits the encoded data to the receiving end via the transmission channel.

該步驟(D)中，在該接收端接收到經該傳輸通道之雜訊干擾的該編碼資料後，該接收端根據所儲存的相關於該生成矩陣的一校驗矩陣獲得一目標症狀值。In the step (D), after the receiving end receives the coded data interfered by the noise of the transmission channel, the receiving end obtains a target symptom value according to a stored check matrix related to the generating matrix.

該步驟(E)中，該接收端根據該查找表判定該目標症狀值是否對應一錯誤樣式。In step (E), the receiving end determines whether the target symptom value corresponds to an error pattern according to the lookup table.

該步驟(F)中，在判定出該目標症狀值對應一錯誤樣式後，該接收端根據該目標症狀值所對應的該錯誤樣式更正經該傳輸通道之雜訊干擾的該編碼資料的錯誤，以獲得該傳輸資料。In step (F), after determining that the target symptom value corresponds to an error pattern, the receiving end corrects the error of the encoded data interfered by the noise of the transmission channel according to the error pattern corresponding to the target symptom value, To obtain the transmission data.

本發明之功效在於：該傳送端產生該生成矩陣進行編碼，藉此，該接收端能根據該目標症狀值及該查找表直接進行解碼。The effect of the present invention is that the transmitting end generates the generating matrix for encoding, whereby the receiving end can directly decode according to the target symptom value and the lookup table.

在本發明被詳細描述前，應當注意在以下的說明內容中，類似的元件是以相同的編號來表示。Before the present invention is described in detail, it should be noted that in the following description, similar elements are denoted by the same number.

參閱圖1，本發明代數幾何碼的赫米特碼之編碼及解碼方法由一系統1實施，該系統1包括一傳送端11及一接收端12。Referring to FIG. 1, the encoding and decoding method of the Hermitian code of the algebraic geometric code of the present invention is implemented by a system 1. The system 1 includes a transmitting end 11 and a receiving end 12.

該傳送端11經由一傳輸通道13傳輸一傳輸資料 m至該接收端12，該傳送端11儲存一參數資訊，其中該參數資訊包括相關於位於一個三維座標系統的一不可約仿射平滑曲線的一參數曲線方程式

Figure 02_image001
、一有限體GF(
Figure 02_image003
)的
Figure 02_image005
個元素，及一預設度數(degree) a。該參數曲線方程式
Figure 02_image006
，xyz表示該三維座標系統的座標軸， r為正整數。 The transmitting end 11 transmits a transmission data m to the receiving end 12 via a transmission channel 13, and the transmitting end 11 stores a parameter information, wherein the parameter information includes an irreducible affine smooth curve related to a three-dimensional coordinate system A parametric curve equation
Figure 02_image001
, A finite body GF (
Figure 02_image003
)of
Figure 02_image005
Elements, and a preset degree (degree) a . The parametric curve equation
Figure 02_image006
, Xyz represents the coordinate axis of the three-dimensional coordinate system, r is a positive integer.

該接收端12儲存有一症狀值(Syndrome)對錯誤樣式(Error Patterns)的查找表，該查找表包括多個症狀值及多個相關於該傳輸通道13之雜訊且分別對應該等症狀值的錯誤樣式。The receiving end 12 stores a lookup table of symdrome values to error patterns. The lookup table includes a plurality of symptom values and a plurality of noises related to the transmission channel 13 corresponding to the symptom values Wrong style.

本發明代數幾何碼的赫米特碼之編碼及解碼方法之一實施例包含一編碼程序及一解碼程序。An embodiment of the encoding and decoding method of the Hermitian code of the algebraic geometric code of the present invention includes an encoding procedure and a decoding procedure.

參閱圖1、2，以下詳述的該實施例的編碼程序之各個步驟。Referring to Figures 1 and 2, the steps of the encoding process of this embodiment are described in detail below.

在步驟S201中，該傳送端11根據該參數資訊的該參數曲線方程式

Figure 02_image001
及該有限體GF(
Figure 02_image003
)的該等元素，獲得一無窮遠點 Q及
Figure 02_image008
個仿射點
Figure 02_image010
。 In step S201, the transmitter 11 formulates the parameter curve equation according to the parameter information
Figure 02_image001
And the limited body GF (
Figure 02_image003
) To obtain an infinity point Q and
Figure 02_image008
Affine point
Figure 02_image010
.

在步驟S202中，該傳送端11根據該參數資訊的該預設度數 a獲得一代數幾何碼( n, k, d*)，在本實施例中，

Figure 02_image012
代表該代數幾何碼之一碼字長度，
Figure 02_image014
代表該傳輸資料 m的長度，
Figure 02_image016
代表該代數幾何碼之一最小距離，
Figure 02_image018
，但不以此為限。 In step S202, the transmitting end 11 obtains a generation geometric code ( n , k , d *) according to the preset degree a of the parameter information. In this embodiment,
Figure 02_image012
Represents the length of one codeword of the algebraic geometric code,
Figure 02_image014
Represents the length of the transmission data m ,
Figure 02_image016
Represents the minimum distance of one of the algebraic geometric codes,
Figure 02_image018
, But not limited to this.

在步驟S203中，該傳送端11根據該代數幾何碼( n, k, d*)及該參數資訊的該預設度數 a獲得一包括該預設度數 a及該無窮遠點 Q的有理函數組 L( aQ)，該有理函數組 L( aQ)包括 k個有理函數，該有理函數組L( aQ)以下式表示：

Figure 02_image020
，0≦ i＜ p，0≦ j， 其中， p表示該不可約仿射平滑曲線的階數，
Figure 02_image022
表示該等 k個有理函數。 In step S203, the transmitting end 11 obtains a rational function group including the preset degree a and the infinity point Q according to the algebraic geometric code ( n , k , d *) and the preset degree a of the parameter information L ( aQ ), the rational function group L ( aQ ) includes k rational functions, and the rational function group L( aQ ) is expressed by the following formula:
Figure 02_image020
, 0≦ i < p , 0≦ j , where p represents the order of the irreducible affine smooth curve,
Figure 02_image022
Represent these k rational functions.

在步驟S204中，該傳送端11根據該有理函數組 L( aQ)及該等

Figure 02_image008
個仿射點
Figure 02_image024
獲得一生成矩陣 G。值得注意的是，該傳送端11將該等
Figure 02_image025
個仿射點
Figure 02_image026
代入該有理函數組L( aQ)中，以獲得該生成矩陣 G，該生成矩陣 G以下式表示：
Figure 02_image028
。 In step S204, the transmitting end 11 is based on the rational function group L ( aQ ) and these
Figure 02_image008
Affine point
Figure 02_image024
A generator matrix G is obtained . It is worth noting that the transmitting end 11 will
Figure 02_image025
Affine point
Figure 02_image026
The substituting group rational function L (aQ) to obtain the generator matrix G, the generator matrix G represented by the following formula:
Figure 02_image028
.

在步驟S205中，該傳送端11根據該傳輸資料 m及該生成矩陣 G產生包括該傳輸資料的一編碼資料 c，其中 c= mG。在本實施例中，該傳送端11先將該生成矩陣 G轉換成系統化形式(systematic form)，再以該系統化形式的生成矩陣 G sys 將該傳輸資料 m進行編碼，以獲得該編碼資料 c。要特別注意的是，使用者可根據該生成矩陣 G獲得一校驗矩陣 H， GH T =0， H T 為該校驗矩陣 H的轉置矩陣，再將該校驗矩陣 H儲存至該接收端12，但不以此為限。 In step S205, the transmitting end 11 generates a coded data c including the transmission data according to the transmission data m and the generation matrix G , where c = mG . In this embodiment, the transmitting end 11 first converts the generating matrix G into a systematic form, and then encodes the transmission data m with the generating matrix G sys of the systematic form to obtain the encoded data c . Of particular note is that the user can obtain a check matrix H based on the generator matrix G, GH T = 0, H T of the parity check matrix H for the transposed matrix, then the check matrix H is stored to the receiving End 12, but not limited to this.

在步驟S206中，該傳送端11經由該傳輸通道13傳送該編碼資料 c至該接收端12。 In step S206, the transmitting end 11 transmits the encoded data c to the receiving end 12 via the transmission channel 13.

以(8,5,3)赫米特碼(Hermitian Codes)為例，該參數資訊的參數曲線方程式

Figure 02_image030
，該有限體GF(
Figure 02_image032
)=
Figure 02_image034
，且該預設度數 a=5。 Taking (8,5,3) Hermitian codes as an example, the parameter curve equation of the parameter information
Figure 02_image030
, The finite body GF(
Figure 02_image032
)=
Figure 02_image034
, And the preset degree a = 5.

該傳送端11所獲得的該無窮遠點 Q=(0,1,0)，該等

Figure 02_image036
個仿射點
Figure 02_image038
如下表一所示，該赫米特碼之碼字長度
Figure 02_image040
，該傳輸資料 m的長度
Figure 02_image042
，該赫米特碼之最小距離
Figure 02_image044
，
Figure 02_image046
。該有理函數組L(5 Q)以下式表示：
Figure 02_image048
， 因為仿射點的 z=1，因此
Figure 02_image050
， 將該等8個仿射點
Figure 02_image038
代入該有理函數組L(5 Q)中，如表二所示，以獲得該生成矩陣 G以下式表示：
Figure 02_image052
， 該系統化形式的生成矩陣 G sys以下式表示：
Figure 02_image054
， 值得注意的是，該系統化形式的生成矩陣 G sys單位為符號(symbol)，若以位元(bit)表示該系統化形式的生成矩陣 G sys如下式：
Figure 02_image056
， 其中， I 10為10×10的單位矩陣(Identity matrix)。
Figure 108104374-A0305-0001
表一
Figure 02_image067
表二 The infinity point Q = (0,1,0) obtained by the transmitting end 11
Figure 02_image036
Affine point
Figure 02_image038
As shown in Table 1 below, the code length of the Hermitian code
Figure 02_image040
, The length of the transmission data m
Figure 02_image042
, The minimum distance of the Hermitian code
Figure 02_image044
,
Figure 02_image046
. The rational function group L(5 Q ) is expressed by the following formula:
Figure 02_image048
, Because z = 1 of the affine point, so
Figure 02_image050
, The 8 affine points
Figure 02_image038
Substitute into the rational function group L(5 Q ), as shown in Table 2, to obtain the generator matrix G expressed by the following formula:
Figure 02_image052
The systematic matrix G sys is expressed by the following formula:
Figure 02_image054
It is worth noting that the unit of the systemized form of the generator matrix G sys is a symbol. If the systemized form of the generator matrix G sys is expressed in bits as follows:
Figure 02_image056
, Where I 10 is a 10×10 identity matrix.
Figure 108104374-A0305-0001
 Table I
Figure 02_image067
Table II

因此，假設該傳輸資料 m=(1011001011)，則該編碼資料 c=(1011001011111100)。 Therefore, assuming that the transmission data m = (1011001011), then the coded data c = (1011001011111100).

參閱圖1、3，以下詳述的該實施例的解碼程序的各個步驟。Referring to FIGS. 1 and 3, each step of the decoding program of this embodiment is described in detail below.

在步驟S301中，在該接收端12接收到一接收資料 r後，該接收端12根據該接收資料 r及所儲存的該校驗矩陣 H獲得一目標症狀值 S 0，其中 S 0 = r H T 。 In step S301, after the receiving end 12 receives a received data r , the receiving end 12 obtains a target symptom value S 0 according to the received data r and the stored check matrix H , where S 0 = r H T.

在步驟S302中，該接收端12判定該目標症狀值 S 0是否為零位元串。當判定出該目標症狀值 S 0 為零位元串時，即認為該接收資料 r為未經該傳輸通道13之雜訊干擾的該編碼資料 c，進行步驟S303；當判定出該目標症狀值 S 0 不為零位元串時，表示該接收資料 r即為經該傳輸通道13之雜訊干擾的該編碼資料

Figure 02_image069
，則進行步驟S304。 In step S302, the receiving end 12 determines whether the target symptom value S 0 is a zero bit string. When it is determined that the target symptom value S 0 is a zero-bit string, the received data r is considered to be the coded data c that is not disturbed by the noise of the transmission channel 13, and step S303 is performed; when the target symptom value is determined When S 0 is not a zero bit string, it means that the received data r is the encoded data interfered by the noise of the transmission channel 13
Figure 02_image069
, Proceed to step S304.

在步驟S303中，該接收端12產生一指示出無錯誤發生不需要更正的無錯誤訊息。In step S303, the receiving end 12 generates an error-free message indicating that no error occurs and no correction is needed.

在步驟S304中，該接收端12根據該查找表判定該目標症狀值 S 0 是否對應一錯誤樣式。當判定出該目標症狀值 S 0對應一錯誤樣式時，進行步驟S305；當判定出該目標症狀值 S 0不對應任何錯誤樣式時，則進行步驟S306。 In step S304, the receiving end 12 determines whether the target symptom value S 0 corresponds to an error pattern according to the look-up table. When it is determined that the target symptom value S 0 corresponds to an error pattern, step S305 is performed; when it is determined that the target symptom value S 0 does not correspond to any error pattern, step S306 is performed.

在步驟S305中，該接收端12根據該目標症狀值所對應的該錯誤樣式更正經該傳輸通道13之雜訊干擾的該編碼資料

Figure 02_image069
的錯誤，以獲得該傳輸資料 m。值得注意的是，在本實施例中，該接收端12將該錯誤樣式與經該傳輸通道13之雜訊干擾的該編碼資料
Figure 02_image069
進行模二加法(Modulo-2 Addition)以更正經該傳輸通道13之雜訊干擾的該編碼資料
Figure 02_image069
的錯誤，而獲得該傳輸資料 m。 In step S305, the receiving end 12 corrects the encoded data interfered by the noise of the transmission channel 13 according to the error pattern corresponding to the target symptom value
Figure 02_image069
Error to obtain the transmission data m . It is worth noting that in this embodiment, the receiving end 12 interferes with the encoded data of the error pattern and the noise through the transmission channel 13
Figure 02_image069
Modulo-2 Addition is performed to correct the coded data interfered by the noise of the transmission channel 13
Figure 02_image069
Error while getting the transmitted data m .

在步驟S306中，該接收端12產生一指示出無法更正的錯誤訊息。In step S306, the receiving end 12 generates an error message indicating that it cannot be corrected.

以(8,5,3)赫米特碼為例，假設經該傳輸通道13之雜訊干擾的該編碼資料=(1101001011111100)時，則該接收端12根據該接收資料 r及所儲存的該校驗矩陣 H獲得該目標症狀值 S 0 =(111111)，該校驗矩陣 H如下式所示：

Figure 02_image072
， 其中， I 6為6×6的單位矩陣。該查找表如表三所示，該接收端12根據該查找表判定出該目標症狀值 S 0 對應一錯誤樣式(0110000000)，經該傳輸通道13之雜訊干擾的該編碼資料
Figure 02_image074
=(1101001011111100)與該目標症狀值 S 0 對應的該錯誤樣式進行模二加法，獲得該編碼資料 c=(1011001011111100)，其中該編碼資料 c的前10位元即為該傳輸資料 m=(1011001011)。 Taking (8, 5, 3) Hermitian code as an example, assuming that the encoded data interfered by the noise of the transmission channel 13 = (1101001011111100), then the receiving end 12 according to the received data r and the stored The check matrix H obtains the target symptom value S 0 =(111111), and the check matrix H is as follows:
Figure 02_image072
, Where I 6 is a 6×6 identity matrix. The look-up table is shown in Table 3. The receiving end 12 determines that the target symptom value S 0 corresponds to an error pattern (0110000000) according to the look-up table, and the encoded data interfered by the noise of the transmission channel 13
Figure 02_image074
=(1101001011111100) The error pattern corresponding to the target symptom value S 0 is modulo two added to obtain the encoded data c = (1011001011111100), where the first 10 bits of the encoded data c are the transmitted data m = (1011001011 ).

假設經該傳輸通道13之雜訊干擾的該編碼資料

Figure 02_image075
=(0001001011111100)時，則該接收端12根據該接收資料 r及所儲存的該校驗矩陣 H獲得該目標症狀值 S 0 =(101101)，由於該目標症狀值 S 0 不對應該查找表的任何錯誤樣式，該接收端12無法更正錯誤，因此該接收端12產生該錯誤訊息。
Figure 108104374-A0305-0002
 Assume that the encoded data interfered by the noise of the transmission channel 13
Figure 02_image075
=(0001001011111100), the receiving end 12 obtains the target symptom value S 0 =(101101) according to the received data r and the stored check matrix H , because the target symptom value S 0 does not correspond to any of the lookup table For the error pattern, the receiving end 12 cannot correct the error, so the receiving end 12 generates the error message.
Figure 108104374-A0305-0002

要特別注意的是，一般而言，碼參數為(n,k,d*)的錯誤更正碼的錯誤更正能力

Figure 02_image076
，其中
Figure 02_image078
為下取整函數(floor function)，上例的錯誤更正能力
Figure 02_image080
，然而若錯誤發生在連續的位元，其錯誤更正能力 t可達到2。 It is important to note that, in general, the error correction capability of the error correction code with the code parameter (n, k, d*)
Figure 02_image076
,among them
Figure 02_image078
To correct the floor function, the error correction capability of the above example
Figure 02_image080
However, if the error occurs in consecutive bits, the error correction capability t can reach 2.

綜上所述，本發明代數幾何碼的赫米特碼之編碼及解碼方法，該傳送端11以該參數資訊產生該生成矩陣 G，以進行快速編碼，藉此，該接收端12能根據該目標症狀值及該查找表直接進行解碼，故確實能達成本發明的目的。 In summary, in the encoding and decoding method of the Hermitian code of the algebraic geometric code of the present invention, the transmitting end 11 generates the generating matrix G based on the parameter information for fast encoding, whereby the receiving end 12 can The target symptom value and the lookup table are directly decoded, so it can indeed achieve the purpose of the invention.

惟以上所述者，僅為本發明的實施例而已，當不能以此限定本發明實施的範圍，凡是依本發明申請專利範圍及專利說明書內容所作的簡單的等效變化與修飾，皆仍屬本發明專利涵蓋的範圍內。However, the above are only examples of the present invention, and the scope of implementation of the present invention cannot be limited by this, any simple equivalent changes and modifications made according to the scope of the patent application of the present invention and the content of the patent specification are still classified as Within the scope of the invention patent.

Figure 108104374-A0305-0003
Figure 108104374-A0305-0003

本發明的其他的特徵及功效，將於參照圖式的實施方式中清楚地呈現，其中： 圖1是一方塊圖，說明一用來實施本發明代數幾何碼的赫米特碼之編碼及解碼方法的一實施例之系統； 圖2是一流程圖，說明該實施例之一編碼程序；及 圖3是一流程圖，說明該實施例之一解碼程序。 Other features and functions of the present invention will be clearly presented in the embodiments with reference to the drawings, in which: 1 is a block diagram illustrating a system for implementing an embodiment of the encoding and decoding method of the Hermitian code of the algebraic geometric code of the present invention; Figure 2 is a flowchart illustrating one of the encoding procedures of this embodiment; and FIG. 3 is a flowchart illustrating a decoding procedure in this embodiment.

S201~S206:步驟 S201~S206: Steps

Claims (6)

一種代數幾何碼之編碼及解碼方法，由一系統所執行，該系統包括一傳送端及一接收端，該傳送端經由一傳輸通道傳輸一傳輸資料至該接收端，該接收端儲存有一症狀值對錯誤樣式的查找表，該查找表包括多個症狀值及多個相關於傳輸通道之雜訊且分別對應該等症狀值的錯誤樣式，該編碼及解碼方法包含以下步驟：(A)藉由該傳送端，根據所儲存的一參數資訊產生一生成矩陣，其中該參數資訊包括相關於位於三維座標系統的一不可約仿射平滑曲線的一參數曲線方程式、一有限體的多個元素，及一預設度數，其中，步驟(A)包括以下子步驟：(A-1)根據該參數資訊的該參數曲線方程式C(x,y,z)及該有限體的該等元素，獲得一無窮遠點Q及多個仿射點，(A-2)根據該參數資訊的該預設度數a獲得一代數幾何碼(n,k,d*)，其中，n代表該代數幾何碼之一碼字長度，k代表該傳輸資料的長度，d*代表該代數幾何碼之一最小距離，(A-3)根據該代數幾何碼(n,k,d*)及該參數資訊的該預設度數a獲得一包括該預設度數a及該無窮遠點Q的有理函數組L(aQ)，該有理函數組L(aQ)包括k個有理函數，及(A-4)根據該有理函數組L(aQ)及該等仿射點獲得該生成矩陣G；(B)藉由該傳送端，根據該傳輸資料及該生成矩陣產 生包括該傳輸資料的一編碼資料；(C)藉由該傳送端，經由該傳輸通道傳送該編碼資料至該接收端；(D)藉由該接收端，在接收到經該傳輸通道之雜訊干擾的該編碼資料後，根據所儲存的相關於該生成矩陣的一校驗矩陣獲得一目標症狀值；(E)藉由該接收端，根據該查找表判定該目標症狀值是否對應一錯誤樣式；及(F)藉由該接收端，在判定出該目標症狀值對應一錯誤樣式後，根據該目標症狀值所對應的該錯誤樣式更正經該傳輸通道之雜訊干擾的該編碼資料的錯誤，以獲得該傳輸資料。 An encoding and decoding method of algebraic geometric codes is executed by a system. The system includes a transmitting end and a receiving end. The transmitting end transmits a transmission data to the receiving end through a transmission channel. The receiving end stores a symptom value For a lookup table of error patterns, the lookup table includes a plurality of symptom values and a plurality of error patterns related to the noise of the transmission channel and corresponding to the symptom values respectively. The encoding and decoding method includes the following steps: (A) The transmitting end generates a generating matrix according to a stored parameter information, wherein the parameter information includes a parameter curve equation related to an irreducible affine smooth curve located in the three-dimensional coordinate system, a plurality of elements of a finite body, and A preset degree, wherein step (A) includes the following sub-steps: (A-1) based on the parameter curve equation C(x,y,z) of the parameter information and the elements of the finite body, obtain an infinity The far point Q and multiple affine points, (A-2) obtain the algebraic geometric code ( n , k , d *) according to the preset degree a of the parameter information, where n represents a code of the algebraic geometric code Word length, k represents the length of the transmitted data, d * represents one of the minimum distance of the algebraic geometric code, (A-3) according to the algebraic geometric code ( n , k , d *) and the preset degree of the parameter information a obtains a rational function group L ( aQ ) including the preset degree a and the infinity point Q , the rational function group L ( aQ ) includes k rational functions, and (A-4) according to the rational function group L ( aQ ) and the affine points obtain the generating matrix G ; (B) by the transmitting end, generating a coded data including the transmitting data according to the transmission data and the generating matrix; (C) by the transmitting end , Transmitting the coded data to the receiving end through the transmission channel; (D) through the receiving end, after receiving the coded data interfered by the noise of the transmission channel, according to the stored correlation with the generation matrix A check matrix obtains a target symptom value; (E) determines whether the target symptom value corresponds to an error pattern based on the look-up table by the receiving end; and (F) determines the target symptom by the receiving end After the value corresponds to an error pattern, the error of the encoded data interfered by the noise of the transmission channel is corrected according to the error pattern corresponding to the target symptom value to obtain the transmission data. 如請求項1所述的代數幾何碼之編碼及解碼方法，其中，在子步驟(A-1)中，該有限體為一具有r 2個元素的有限體GF(r 2)，該參數曲線方程式C(x,y,z)=xr+1+y r z+yz r ，該等仿射點為r 3個，其中xyz表示該三維座標系統的座標軸，r為正整數。 The encoding and decoding method of an algebraic geometric code according to claim 1, wherein, in the sub-step (A-1), the finite volume is a finite volume GF( r 2 ) with r 2 elements, and the parameter curve Equation C(x,y,z)=x r+1 + y r z + yz r , the affine points are r 3 , where xyz represents the coordinate axis of the three-dimensional coordinate system, and r is a positive integer. 如請求項2所述的代數幾何碼之編碼及解碼方法，其中，在子步驟(A-2)中，其中，該碼字長度n=r 3，該傳輸資料的長度k=n-a+γ-1，該最小距離d*=a-2γ+2，
Figure 108104374-A0305-02-0014-1
 The encoding and decoding method of an algebraic geometric code as described in claim 2, wherein, in sub-step (A-2), wherein the codeword length n = r 3 and the transmission data length k = n - a + γ -1, the minimum distance d *= a -2 γ +2,
Figure 108104374-A0305-02-0014-1
 如請求項3所述的代數幾何碼之編碼及解碼方法，其中，在子步驟(A-3)中，該有理函數組L(aQ)以下式表示：
Figure 108104374-A0305-02-0015-2
 其中，m表示該不可約仿射平滑曲線的階數，f 1,f 2,…,f k 表示該等k個有理函數。 The encoding and decoding method of an algebraic geometric code as described in claim 3, wherein in the sub-step (A-3), the rational function group L ( aQ ) is expressed by the following formula:
Figure 108104374-A0305-02-0015-2
 Where m represents the order of the irreducible affine smooth curve, and f 1 , f 2 ,..., f k represent the k rational functions. 如請求項4所述的代數幾何碼之編碼及解碼方法，其中，在子步驟(A-4)中，將該等r 3個仿射點代入該有理函數組L(aQ)中，以獲得該生成矩陣G，該生成矩陣G以下式表示：
Figure 108104374-A0305-02-0015-4
 其中P 1,P 2,…,
Figure 108104374-A0305-02-0015-6
表示該等r 3個仿射點。 The encoding and decoding method of the algebraic geometric code as described in claim 4, wherein, in sub-step (A-4), the r 3 affine points are substituted into the rational function group L ( aQ ) to obtain the generator matrix G, the generator matrix G represented by the following formula:
Figure 108104374-A0305-02-0015-4
 Where P 1 , P 2 ,...,
Figure 108104374-A0305-02-0015-6
Represent these r 3 affine points. 如請求項1所述的代數幾何碼之編碼及解碼方法，在步驟(E)還包含以下步驟：(G)藉由該接收端，在判定出已根據該查找表的該目標症狀值不對應任何錯誤樣式後，產生一指示出無法更正的錯誤訊息。 According to the encoding and decoding method of the algebraic geometric code described in claim 1, step (E) further includes the following steps: (G) With the receiving end, it is determined that the target symptom value according to the lookup table does not correspond After any error pattern, an error message indicating that it cannot be corrected is generated.
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