Mac Williams Extension Theorem for Codes in Projective Modules over a Finite Frobenius Ring

Abstract

Let C be a right submodule of a free finite right module F over a finite Frobenius ring R. In order to facilitate the decoding of C, one can replace F by the minimal projective submodule P of F, containing C. Once R is Frobenius, P coincides with the injective envelope E(C) of C in F and can be constructed as the maximal essential extension of C in F. The work describes the injective envelope E(C) of an arbitrary module C over a residue ring Zm modulo a natural number m. Any choice of a basis of F is associated with an inner product on F and allows to describe the relations of C in F by its left orthogonal group (⊥C, +) < (F, +). Bearing in mind that the relations (⊥P, +) of P = E(C) in F are contained in (⊥C, +), one realizes the relations of C in P by the quotient group (⊥C, +)/(⊥P, +). The decoding of C in P = E(C) makes use of a generating set of (⊥C, +)/(⊥P, +), which is obviously smaller than a generating set of (⊥C, +), needed for a decoding of C in F.The present note discusses also Mac Williams Extension Theorem for codes C in finite right projective modules P over a finite Frobenius ring R. It establishes that any Hamming isometry of C in P lifts to a monomial transformation g of a free cover F of P. Moreover, if the kernel of the epimorphism F onto P is invariant under g then the Hamming isometry of C in P extends to a Hamming isometry of P with itself.