Double circulant quadratic residue codes

We give a lower bound for the minimum distance of double circulant binary quadratic residue codes for primes p≡±3(mod8). This bound improves on the square root bound obtained by Calderbank and Beenker, using a completely different technique. The key to our estimates is to apply a result by Helleseth, to which we give a new and shorter proof. Combining this result with the Weil bound leads to the improvement of the Calderbank and Beenker bound. For large primes p, their bound is of order √(2p) while our new improved bound is of order 2√p. The results can be extended to any prime power q and the modifications of the proofs are briefly indicated.