ON REPEATED-ROOT CYCLIC CODES

A parity-check matrix for a q-ary repeated-root cyclic code is derived using the Hasse derivative. Then the minimum distance of a q-ary repeated-root cycli code C is expressed in terms of the minimum distance of a certain simple-root cyclic code CBAR that is determined by C. With the help of this result, several binary repeated-root cyclic codes of lengths up to n = 62 are shown to contain the largest known number of codewords for their given length and minimum distance. It is further shown that to a q-ary repeated-root cyclic code C of length n = p-delta-nBAR, where p is the characteristic of GF(q) and gcd(p,nBAR) = 1, there corresponds a simple-root cyclic code C tripple-over-dot of rate and relative minimum distance at least as large as the corresponding values of C, however, of length nBAR, i.e., shorter by a factor of p-delta. The relative minimum distance d(min)/n of q-ary repeated-root cyclic codes C of rate r greater-than-or-equal-to R is proven to tend to zero as the largest multiplicity of a root of the generator g(x) increases to infinity. It is further shown that repeated-root cyclic codes cannot be asymptotically better than simple-root cyclic codes.