DOUBLE-ERROR-CORRECTING CYCLIC CODES AND ABSOLUTELY IRREDUCIBLE POLYNOMIALS OVER GF(2)

Codewords of weight less than or equal to 4 in certain cyclic codes of length n = 2(s) - 1, parameterized by an odd integer t, can be relate d to zeros of certain projective plane curves g(t)(X, Y, Z). Some fami lies of these codes have been shown to have no codewords of weight les s than or equal to 4, i.e., they are 2-error-correcting codes. But if the polynomials g(t)(X, Y, Z) are absolutely irreducible, Well's theor em 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 g (t)(X, Y, Z) is, absolutely irreducible for all integers t > 3 such th at t = 3(mod4), and also for some values t = 1(mod4). These cases prov ide us with evidence for a conjecture that would classify all such cod es in terms of their minimum distance. The methods of proving absolute irreducibility involve Bezout's theorem and may be of independent int erest. (C) 1995 Academic Press, Inc.