Lexicographic code

Lexicographic codes or lexicodes are greedily generated error-correcting codes with remarkably good properties. They were produced independently by Vladimir Levenshtein ^[1] and by John Horton Conway and Neil Sloane. ^[2] The binary lexicographic codes are linear codes, and include the Hamming codes and the binary Golay codes. ^[2]

Construction

A lexicode of length n and minimum distance d over a finite field is generated by starting with the all-zero vector and iteratively adding the next vector (in lexicographic order) of minimum Hamming distance d from the vectors added so far. As an example, the length-3 lexicode of minimum distance 2 would consist of the vectors marked by an "X" in the following example:

Vector	In code?
000	X
001
010
011	X
100
101	X
110	X
111

Here is a table of all n-bit lexicode by d-bit minimal hamming distance, resulting of maximum 2^m codewords dictionary. For example, F₄ 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.

n \ d	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
1	1
2	2	1
3	3	2	1
4	4	3	1	1
5	5	4	2	1	1
6	6	5	3	2	1	1
7	7	6	4	3	1	1	1
8	8	7	4	4	2	1	1	1
9	9	8	5	4	2	2	1	1	1
10	10	9	6	5	3	2	1	1	1	1
11	11	10	7	6	4	3	2	1	1	1	1
12	12	11	8	7	4	4	2	2	1	1	1	1
13	13	12	9	8	5	4	3	2	1	1	1	1	1
14	14	13	10	9	6	5	4	3	2	1	1	1	1	1
15	15	14	11	10	7	6	5	4	2	2	1	1	1	1	1
16	16	15	11	11	8	7	5	5	2	2	1	1	1	1	1	1
17	17	16	12	11	9	8	6	5	3	2	2	1	1	1	1	1	1
18	18	17	13	12	9	9	7	6	3	3	2	2	1	1	1	1	1	1
19	19	18	14	13	10	9	8	7	4	3	2	2	1	1	1	1	1	1
20	20	19	15	14	11	10	9	8	5	4	3	2	2	1	1	1	1	1
21	21	20	16	15	12	11	10	9	5	5	3	3	2	2	1	1	1	1
22	22	21	17	16	12	12	11	10	6	5	4	3	2	2	1	1	1	1
23	23	22	18	17	13	12	12	11	6	6	5	4	2	2	2	1	1	1
24	24	23	19	18	14	13	12	12	7	6	5	5	3	2	2	2	1	1
25	25	24	20	19	15	14	12	12	8	7	6	5	3	3	2	2	1	1
26	26	25	21	20	16	15	12	12	9	8	7	6	4	3	2	2	2	1
27	27	26	22	21	17	16	13	12	9	9	7	7	5	4	3	2	2	2
28	28	27	23	22	18	17	13	13	10	9	8	7	5	5	3	3	2	2
29	29	28	24	23	19	18	14	13	11	10	8	8	6	5	4	3	2	2
30	30	29	25	24	19	19	15	14	12	11	9	8	6	6	5	4	2	2
31	31	30	26	25	20	19	16	15	12	12	10	9	6	6	6	5	3	2
32	32	31	26	26	21	20	16	16	13	12	11	10	7	6	6	6	3	3
33	...	32	...	26	...	21	...	16	...	13	...	11	...	7	...	6	...	3

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. ^[3]

Implementation

Following C generate lexicographic code and parameters are set for the Golay code (N=24, D=8).

#include<stdio.h>#include<stdlib.h>intmain(){/* GOLAY CODE generation */inti,j,k;int_pc[1<<16]={0};// PopCount Macrofor(i=0;i<(1<<16);i++)for(j=0;j<16;j++)_pc[i]+=(i>>j)&1;#define pc(X) (_pc[(X)&0xffff] + _pc[((X)>>16)&0xffff])#define N 24 // N bits#define D 8  // D bits distanceunsignedint*z=malloc(1<<29);for(i=j=0;i<(1<<N);i++){// Scan all previousfor(k=j-1;k>=0;k--)// lexicodes.if(pc(z[k]^i)<D)// Reverse checkingbreak;// is way faster...if(k==-1){// Add new lexicodefor(k=0;k<N;k++)// & print itprintf("%d",(i>>k)&1);printf(" : %d\n",j);z[j++]=i;}}}

Combinatorial game theory

The theory of lexicographic codes is closely connected to combinatorial game theory. In particular, the codewords in a binary lexicographic code of distance d encode the winning positions in a variant of Grundy's game, played on a collection of heaps of stones, in which each move consists of replacing any one heap by at most d− 1 smaller heaps, and the goal is to take the last stone. ^[2]

Notes

1. Levenšteĭn, V. I. (1960), "Об одном классе систематических кодов" [A class of systematic codes], Doklady Akademii Nauk SSSR (in Russian), 131 (5): 1011–1014, MR   0122629 ; English translation in Soviet Math. Doklady 1 (1960), 368–371
2. Conway, John H.; Sloane, N. J. A. (1986), "Lexicographic codes: error-correcting codes from game theory", IEEE Transactions on Information Theory , 32 (3): 337–348, CiteSeerX   10.1.1.392.795 , doi:10.1109/TIT.1986.1057187, MR   0838197
3. Trachtenberg, Ari (2002), "Designing lexicographic codes with a given trellis complexity", IEEE Transactions on Information Theory , 48 (1): 89–100, doi:10.1109/18.971740, MR   1866958

External links

• Bob Jenkins table of binary lexicodes
• On-line generator for lexicodes and their variants
• OEIS sequenceA075928(List of codewords in binary lexicode with Hamming distance 4 written as decimal numbers.)
• Error-Correcting Codes on Graphs: Lexicodes, Trellises and Factor Graphs