On the minimum distances of non-binary cyclic codes
Cyclic codes over \(\text{GF}(q)\) of length \(q^m-1\) are studied. The codes have generator polynomials \(g(x)\) which are products of two distinct minimal polynomials. Let \(\gamma\) be an element of order \(q^m-1\) and let \(g(x)\) be the product of the minimal polynomials of \(\gamma^i\) and \(\gamma^j\). Let \(d_{ij}\) be the minimum distance of this cyclic code. The authors first show that \(d_{i,j}=2\) if and only if \(\text{gcd}(i,j,q^m-1)>1\) or \(\text{gcd}(j-i,q-1)>1\). Further, if \(q=3\) and neither of these conditions hold then \(d_{i,j}=3\) or 4 and both these values occur. If \(q>3\) and neither of these conditions hold then \(d_{i,j}=3\). In the special case \(q=3\) and \(t=3^s+1\), then \(d_{1,t}=3\) if \(m/\text{gcd}(m,s)\) is even and \(d_{1,t}=4\) if \(m/\text{gcd}(m,s)\) is odd.
scientific article; zbMATH DE number 1392727