Let$p$be a prime such that$p\equiv -1\pmod {8}$. Let$k=(p+1)/2$and write$k=2^{m}q$,$q$odd. Let$S=F_{2}[x]/\langle 1+x^{k}\rangle $where$F_{2}$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$\sigma $the algebra of automorphisms on$S$that sends$X^{i}$to$X^{-i}$. Factor$1+x^{q}$over$F_{2}$into irreducible factors. If the class of those factors in$S$is fixed by$\sigma $up to a unit, then the codes have a double circulant presentation.