The Category of Linear Codes

| In the late 50s Slepian 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 canonical form for the generator matrix of an indecompos-able code so that he might read oo the properties of the code from such a matrix, but such a program proved impossible. We here work over an arbitrary eld and deene a restricted class of inde-composable codes | which we call critical. For these codes there is a quasi-canonical form for the generator matrix. Every inde-composable code has a generator matrix that is obtained from the generator matrix of a critical, indecomposable code by augmentation. As an application of our structure theory we illuminate the perfect linear codes, giving, for example, a \canonical" form for the generator matrix of the ternary Golay code.