The Type of a Self-Dual Code

Abstract

To motivate these initial definitions, we begin by remarking that in the classical theory (cf. van Lint [350], MacWilliams and Sloane [361], Pless, Hu.man and Brualdi [427], Rains and Sloane [454]) a linear error-correcting code C is a subspace of a vector space V over a finite field F, with inner products of codewords taking values in F itself. The classical theory was enlarged in the early 1990’s by the discovery by Hammons, Kumar, Calderbank, Sloane and Solé [175], [91], [227] that certain notorious nonlinear binary codes (the Nordstrom-Robinson, Kerdock and Preparata codes) could best be understood as arising from linear codes over the ring Z/4Z, and, in the case of the Kerdock code, from a self-dual linear code over Z/4Z.