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 min imum distance of a q-ary repeated-root cyclic code C is expressecin terms of the min imum distance of a certain simple-root cyclic code C 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 min imum distance. It is further shown that to a q-ary repeated-root cyclic code C of length n = p%, where p is the characteristic of GF(q) and gcd(p,?i) = 1, there corre. sponds a simple-root cyclic code C of rate and relative min imum distance at least as large as the corresponding values of C, however, of length ii, i.e., shorter by a factor of p’. The relative min imum distance dmin /n of q-ary repeated-root cyclic codes C of rate r 2 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. Mar Terms -Cyclic codes, generator polynomial, formal derivative, Hasse derivative.