Quadratic residue codes in their prime
The standard methods for constructing quadratic residue codes require that the characteristic \(p\) of the base field \(GF(q)\) from which they are obtained be invertible in the alphabet ring. However, the spanning sets created do not show this, and they can be read into a field of characteristic \(p\). The resulting codes are invariant under a monomial action of \(GL(2,q)\), rather than one just of \(SL(2,q)\), as for the conventional codes. The invariant subspaces of the action can be produced; the lattice of these additional codes turns out to depend only on the exponent in \(q\) as a power of \(p\). This paper describes the codes and investigates their dimensions and minimum weights. One of them is an extremal type III code of length 28, and certain others are reminiscent of Reed-Muller codes.
scientific article; zbMATH DE number 63896