On Cyclic Codes of Composite Length and the Minimum Distance II

In this paper, we provide two complementary results for cyclic codes of composite length. First, we give a general construction of cyclic codes of length <inline-formula> <tex-math notation="LaTeX">$nr$ </tex-math></inline-formula> from cyclic codes of length <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> and a lower bound on the minimum distance. Numerical data show that many cyclic codes of composite length with the best parameters can be obtained in this way. Second, in the other direction, we show that for cyclic codes of length <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> with a primitive <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-th root of unity as a non-zero, the minimum distance is roughly bounded by the square-free part of <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>. This means that we shall not expect good cyclic codes of length <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> in general if, for example, the length <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> is divisible by a high power of a prime.