Self-dual Codes Using Image Restoration Techniques

Abstract

From past literature it is evident that the search for self-dual codes has been hampered by the computational difficulty of generating the Hadamard matrices required. The use of the cocyclic construction of Hadamard matrices has permitted substantial cut-downs in the search time, but the search space still grows exponentially. Here we look at an adaptation of image-processing techniques for the restoration of damaged images for the purpose of sampling the search space systematically. The performance of this approach is evaluated for Hadamard matrices of small orders, where a full search is possible.

The dihedral cocyclic Hadamard matrices obtained by this technique are used in the search for self-dual codes of length 40,56 and 72. In addition to the extremal doubly-even [56,28,12] code, and two singlyeven [56,28,10] codes, we found a large collection of codes with only one codeword of minimum length.