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IEEE Transactions on Information Theory

Nine good rate (m-1)/pm quasi-cyclic codes

Thomas Aaron Gulliver, V.K. Bhargava

Published Jul 1, 1992• 41 citations
Publisher

Abstract

The class of quasi-cyclic (QC) codes has been proven to contain many good codes. To date the known results are largely codes of the form 1/p and (p-1)/p constructed from circulant matrices. A generalization of these rate 1/p codes to rate (m-1)/pm codes based on the theory of 1-generator QC codes is presented. The results of a search for good codes based on heuristic combinatorial optimization are nine codes which improve the known lower bounds on the minimum distance of binary linear codes.>

DOI
10.1109/18.144718

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The class of quasi-cyclic (QC) codes has been proven to contain many good codes. To date the known results are largely codes of the form 1/p and (p-1)/p constructed from circulant matrices. A generalization of these rate 1/p codes to rate (m-1)/pm codes based on the theory of 1-generator QC codes is presented. The results of a search for good codes based on heuristic combinatorial optimization are nine codes which improve the known lower bounds on the minimum distance of binary linear codes.>
In this paper we present a method to search q circulant matrices; the concatenation of these circulant matrices with circulant identity matrix generates quasi-cyclic codes with high various code rate q/(q+1) (q an integer). This method searches circulant matrices in order to find the good quasi-cyclic code (QCC) having the largest minimum distance. A modified simulated annealing algorithm is used as an evaluator tool of the minimum distance of the obtained QCC codes. Based on this method we found 16 good quasi-cyclic codes with rates (1/2, 2/3 and 3/4), their estimated minimum distance reaches the lower bounds of codes considered to be the better linear block codes in Brouwer’s database.
It has been well-known that the class of quasi-cyclic (QC) codes contain many good codes. In this paper, a method to conduct a computer search for binary 222-generator QC codes is presented, and a large number of good 222-generator QC codes have been obtained. 555 new binary QC codes that improve the lower bounds on minimum distance are presented. Furthermore, with new 222-generator QC codes and Construction X, 222 new improved binary linear codes are obtained. With the standard construction techniques, another 161616 new binary linear codes that improve the lower bound on the minimum distance have also been obtained. Received: 9 November 2016 Accepted: 22 November 2017
A Random Local Matroid Search Algorithm to Construct Good Rate 1/p1/p1/p Systematic Binary Quasi-Cyclic Codes