Double-Error-Correcting Cyclic Codes and Absolutely Irreducible Polynomials over GF(2)

Abstract

Codewords of weight ≤ 4 in certain cyclic codes of length n = 2s − 1, parameterized by an odd integer t, can be related to zeros of certain projective plane curves gt(X, Y, Z). Some families of these codes have been shown to have no codewords of weight ≤ 4, i.e., they are 2-error-correcting codes. But if the polynomials gt(X, Y, Z) are absolutely irreducible, Weil′s theorem shows that the codes do have codewords of weight 4 for all integers s that are sufficiently large with respect to t. Here we prove that gt(X, Y, Z) is, absolutely irreducible for all integers t > 3 such that t ≡ 3(mod 4), and also for some values t ≡ 1(mod 4). These cases provide us with evidence for a conjecture that would classify all such codes in terms of their minimum distance. The methods of proving absolute irreducibility involve Bezout′s theorem and may be of independent interest.