A double circulant presentation for quadratic residue codes (Corresp.)

For
p \eqiv \pm 1 \pmod{8}
there are two binary codes,
Q(p)
and
N(p)
, each an extended quadratic residue code of length
p+1
and dimension
(p+1)/2
. The existence of double circulant generator matrices for these codes is investigated. A possibly infinite family of primes
p
is presented for which
Q(p)
and
N(p)
must have double circulant generator matrices. Two counterexamples prove the construction is not always possible.