Primitive idempotents and generator polynomials of some minimal cyclic codes of length \({p^n}{q^m}\)
Summary: Let \(p, q\) and \(l\) be distinct odd primes such that \(l\) is a primitive root modulo \(p^n\) as well as modulo \(q^m\) with \(\gcd(\varphi(p),\varphi(q)) = 2\). Then the explicit expressions for the complete set of \(2mn + m + n + 1\) primitive idempotents of the minimal cyclic codes of length \({p^n}{q^m}\) over \(\mathrm{GF}(l)\) are obtained. An algorithm is also given to factorise the polynomial \((x^n - 1)\) over \(\mathrm{GF}(k)\), where \(n\) is an integer such that \(\gcd(n, k) = 1\). Using the algorithm generator polynomials of the above minimal cyclic codes can be computed numerically. Some bounds on the minimum distance of these minimal cyclic codes are also obtained.
scientific article; zbMATH DE number 6503839