Lexicographic code: Difference between revisions

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| title = Lexicographic codes: error-correcting codes from game theory

| volume = 32

| year = 1986| citeseerx = 10.1.1.392.795
}}</ref> The binary lexicographic codes are [[linear code]]s, and include the [[Hamming code]]s and the [[binary Golay code]]s.<ref name=conslo/>

== Construction ==

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

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.

:{| class="wikitable"

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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 (linear algebra) | basis]].<ref>{{citation

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#define N 24 // N bits

#define D 8 // D bits distance

unsigned int * z = malloc(1<<2829);

for (i=j=0; i < (1<<N); i++)

{ // Scan all previous

for (k=j-1; k >= 0; k--) // lexicodes.

if( (pc( z[k]^i ) < D) // Reverse chekingchecking

break; // is way faster...