On the asymptotic badness of cyclic codes with block-lengths composed from a fixed set of prime factors

Abstract

q-ary cyclic codes of rate r≥R and with blocklength $n = p_1^{\alpha _1 } \ldots p_s^{\alpha _s }$ composed from a fixed finite set of primes S={p₁, ..., p _s } are shown to have a minimum distance d _min which is upper bounded by a function d(S,R). Hence, cyclic codes with blocklengths such that all prime factors are in S and which have rate r≥R are asymptotically bad in the sense that for these codes d _min /n tends to zero as n increases to infinity.