The covering radius of cyclic codes of length up to 31

Remark 4: The author is unable to deal analytically with the general case of p + l/2 where one does not have the property of symmetry. However, the case that p is close to l/2 may be tractabie an9 interesting. Linear smoothing using measurements containing correlated noise with an application to inertial navigation, " IEEE Trans. Abstract-The covering radius is given for all binary cyclic codes of length less th?n or equal to 31. Many of these codes are optimal in the sense ?f having the smallest possible covering radius of any linear code of that length and dimension. There has been considerable interest recently in the covering radius of codes (see [l], [3]-[5]), but many open questions remain concerning the covering radius of particular families of codes. In this cofrespondence we give the covering radius R for all cyclic codes of length n I 31. As our source for these codes we used C. L. Chen's table in Peterson and Weldon 16, App. D] with four omissions corrected. Several methods were used to compute R. Most codes were handled by computer. Let C be an [n, k] code. Method 1: By definition, R 5 R, if and only if every vector of weight R, + 1 is within distance R, of some codeword. This method may be implemented by first making a list of the code-words, and then testing each n-tuple of weight R, + 1 to see if it is within distance R, of some codeword. The number of steps is proportional to (1 R,: 1. 2k and 2k words are needed to store the code. Method 2: Let H be a parity check matrix for C. Then C has R I R, if and only if every (n-k)-tuple is the sum of at most R, columns of H [3, Sec. I-A]. This may be implemented by Manuscript received October 26, 1984.