Formula omitted]-extension of linear codes

Abstract

We construct new linear codes with high minimum distance d . In at least 12 cases these codes improve the minimum distance of the previously known best linear codes for fixed parameters n , k . Among these new codes there is an optimal ternary [ 88 , 8 , 54 ] 3 code. We develop an algorithm, which starts with already good codes C , i.e. codes with high minimum distance d for given length n and dimension k over the field G F ( q ) . The algorithm is based on the newly defined ( l , s ) -extension. This is a generalization of the well-known method of adding a parity bit in the case of a binary linear code of odd minimum weight. ( l , s ) -extension tries to extend the generator matrix of C by adding l columns with the property that at least s of the l letters added to each of the codewords of minimum weight in C are different from 0 . If one finds such columns the minimum distance of the extended code is d + s provided that the second smallest weight in C was at least d + s . The question whether such columns exist can be settled using a Diophantine system of equations.