Quadratic residue codes and symplectic groups

Abstract

The extended quadratic residue codes are known to be invariant under a monomial action by the projective special linear group, an action whose permutation part is the ordinary action on the projective line [3, Theorem 3.11 (see also [7], [15], [16]). Th e re p resentations of the special linear group that arise from the codes are among those constructed in [20] (the group coinciding with the symplectic group in the two-dimensional case). It was thought that the algebraic framework used in [20] to produce these representations could also serve as a starting-point for the codes, thus giving the codes and the group action at the same time. The purpose of the present paper is to substantiate that thought. Here is an outline of the paper and its main results: let V be a vector space of even dimension 2n over the finite field GF(p), 4 a power of an odd prime, and let V be endowed with a non-degenerate symplectic (skewsymmetric) form. Let Q(V) be the corresponding symplectic group on V. An algebra A was constructed in [20] that is basically a twisted group algebra of V over a suitable ring, and from A certain representations of Sp( V) were obtained. Section 1 summarizes these results. In Section 2 several functions arising in that development are calculated more explicitly. Their values involve the constant p of 2.1: p2 = 6q, with Q E 6 (mod 4), 6 = 51. To each maximal isotropic subspace of V corresponds a distinct idempotent of A, and the set I of these idempotents is central to the formation of the codes. The multiplication of these idempotents is also considered in Section 2, the most frequently used result being the Lemma 2.3. (2.6 deals with the characters of the representations.) If the symplectic form is scaled by a non-square of GF@), one obtains another algebra A’, and its relation to A is the subject of Section 3. Section 4 contains the construction of the codes and a monomial action