Lexicographic code
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Lexicographic codes or lexicodes are a group of error-correcting codes produced independently by Vladimir Levenshtein and by John Horton Conway and Neil Sloane in 1960 and 1986, respectively. The binary lexicographic codes are linear codes, and include the Hamming codes and the binary Golay codes.
Lexicographic codes or lexicodes are a group of error-correcting codes produced independently by Vladimir Levenshtein and by John Horton Conway and Neil Sloane in 1960 and 1986, respectively. The binary lexicographic codes are linear codes, and include the Hamming codes and the binary Golay codes. For example, F 4 code (n=4,d=2,m=3), extended Hamming code (n=8,d=4,m=4) and especially Golay code (n=24,d=8,m=12) shows exceptional compactness compared to neighbors. All odd d-bit lexicode distances are exact copies of the even d+1 bit distances minus the last dimension, so an odd-dimensional space can never create something new or more interesting than the d+1 even-dimensional space above. Since lexicodes are linear, they can also be constructed by means of their basis. Following C generate lexicographic code and parameters are set for the Golay code (N=24, D=8). include include int main() { /* GOLAY CODE generation */ int i, j, k; int _pc[1 >j)&1; define pc(X) (_pc[(X)&0xffff] + _pc[((X)>>16)&0xffff]) define N 24 // N bits define D 8 // D bits distance unsigned int * z = malloc(1 = 0; k--) // lexicodes. if (pc(z[k]^i) >k)&1); printf(" : %d\n", j); z[j++] = i; } } } Error-Correcting Codes on Graphs: Lexicodes, Trellises and Factor Graphs
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