On Some Double Circulant Binary Extended Quadratic Residue Codes

Let p be a prime such that p = - 1 (mod 8). Let k = (p + 1)/2 and write k = 2^mq, q odd. Let S -F₂[x]/(1 + x^k) where F₂ is the Galois field of two elements. We prove that the binary extended quadratic residue codes of length 2k have a double circulant presentation in the following cases: (1) q = 1, and (2) let X be the class of x in S, and alpha the algebra of automorphisms on S that sends X´ to X^-1. Factor 1 + x^q over F₂ into irreducible factors. If the class of those factors in S is fixed by delta up to a unit, then the codes have a double circulant presentation.