Double-error-correcting cyclic codes and absolutely irreducible polynomials over \(\text{GF}(2)\)
The paper studies the minimum distance of binary cyclic codes of length \(n = 2^s - 1\) with generator polynomial \(m_1 m_t (x)\) when \(m_i (x)\) is the minimal polynomial of \(\omega^i\) over \(GF(2)\) and \(\omega\) is a primitive element of \(GF (2^s)\). Using Weil's theorem and Bezout's theorem, numerical conditions are derived for the minimum distance of such a code to be greater than 4. A conjecture that would classify all such codes in terms of their minimum distance is formulated.
scientific article; zbMATH DE number 840724