New families of error-correcting codes generated by modification of other linear binary block codes

Abstract

Two procedures for modifying linear block error-correction codes are proposed. The first is based on the deletion of x rows and y columns from the parity-check matrix of a given (n, k) code, in such a way that the minimum Hamming distance of the resulting (n−y, k+x−y) code remains equal to that of the original. It is shown that if x is unity, then y may be as low asthe minimum Hamming distance of the (n, n−k) dual of the original (n, k) code. This procedure of deleting rows and columns when applied to the known linear binary block code yields a family of codes, some of which have better rates than those of the best previously known codes of identical Hamming distance and the same number of paritycheck digits. 17 examples of such new codes are derived and included in the paper. It is also shown that apart from appropriate slight modification, the coding and decoding algorithms for this family of modified codes are similar to those of the original code. The second proposed procedure of code modification entails lengthening the original (n, k) linear block code by annexing k′ message digits. If the original code is capable of correcting t random errors or less, then the resulting modified (n+k′,k+k′) code has a rate higher than that of the original (n, k) code. Moreover, its error-correcting capability is such that it will correct t random errors or less if at least one of these occurs in the block of k message digits and s random errors or less, where 1≤s<t, if none of the errors occur in any of the k message digits. Five examples of such codes are given.