Quadratic Residue Codes and Divisibility

Quadratic residue codes are some of the brightest gems of coding theory. Their facets reeect many topics: quadratic residues and reciprocity; multiply transitive and sporadic simple groups; lattices; group representations; and block designs and Hadamard matrices. They provide the only perfect codes for prime-power sized alphabets correcting more than one error, and some are prototypes of extremal self-dual codes with word weights having a common divisor other than one. Andrew Gleason was the rst to deene quadratic residue codes, and his innuence on their development is pervasive. The original codes were cyclic of prime length, along with one-digit extensions. These were then generalized by several people in the 1970's to codes of prime power length and their extensions. A natural framework for them is the class of group algebra codes, codes that are ideals in the group algebra of a nite group. Sections 1 to 4 of this chapter provide an introduction to such codes, including a development of the character theory needed for their discussion when the group is Abelian. Character theory for Abelian groups is also presented in Chapter (Honkala{ Tiett avv ainen) in the development of Gaussian sums and applications to bounds and perfect codes. Sections 5 to 11 form an exposition of the generalized quadratic residue codes covering their main features. These include restrictions on the alphabet eld; duality properties; the extended codes and group actions (the Gleason{Prange theorem); the square-root bound on minimum weights; global codes; and designs. The setting of the geometric codes of Philippe Delsarte, one of the generalizers, is given in Section 9. Quadratic residue codes over small alphabets tend to be divisible: their word weights have a nontrivial common divisor. This divisibility is a consequence of the duality properties, by and large; but much larger divisors occur in generalized Reed-Muller codes. Sections 12 to 16 give an introduction to some of the ideas involved in the study of divisibility. The Gleason-Pierce theorem of Section 13 provides the link between duality and divisibility, and the theorem of Ax in Section 14 explains divisibility in Reed-Muller codes. Generalizations of Ax's theorem are discussed in Section 15. In Section 16, a character formula for the weight function is applied to another divisibility result and a theorem of MacWilliams.