The category of linear codes

Abstract

Slepian (1960) introduced a structure theory for linear, binary codes and proved that every such code was uniquely the sum of indecomposable codes. He had hoped to produce a form for the generator matrix of an indecomposable code so that he might read off the properties of the code from such a matrix, but such a program proved impossible. We here work over an arbitrary field and define a restricted class of indecomposable codes-which we call critical. For these codes there is a quasicanonical form for the generator matrix. Every indecomposable code has a generator matrix that is obtained from the generator matrix of a critical, indecomposable code by augmentation. As an application of the this quasicanonical form we illuminate the perfect linear codes, giving, for example, a canonical form for the generator matrix of the ternary Golay (1949) code.