Primitive idempotents and generator polynomials of some minimal cyclic codes of length \({p^n}{q^m}\) (Q887827)

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.