Covering radius computations for binary cyclic codes

Abstract:

We compute the covering radius of each binary cyclic code of length $\leq 64$ (for both even and odd lengths) and redundancy $\leq 28$. We also compute the covering radii of their punctured codes and shortened codes. Thus we give exact covering radii of over six thousand codes. For each of these codes (except for certain composite codes), we also determine the number of cosets of each weight less than or equal to the covering radius. These results are used to compute the minimum distances of the above cyclic codes. We use the covering radii of shortened codes and other criteria for normality to show that all but eight of the cyclic codes for which we determine the covering radius are normal. For all but seven of these normal codes, we determine the norm using some old results and some new results proved here. We observe that many cyclic codes are among the best covering codes discovered so far, and some of them lead to improvements on the previously published bounds on $t[n,k]$, the smallest covering radius of any binary linear [n, k] code. Among some other applications of our results, we use our table of covering radii and a code augmentation argument to give four improvements on the values of ${d_{\max }}(n,k)$, where ${d_{\max }}(n,k)$ is the largest minimum distance of any binary [n, k] code. These results show that the covering radius is intimately connected with the other three parameters of a linear code, n, k, and d. We also give a complete classification (up to isomorphism) of cyclic self-dual codes of lengths 42, 56, and 60. The computations were carried out mainly on concurrent machines (hypercubes and Connection Machines); we give a description of our algorithm.