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 canonical 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.

A theorem for a binary code arising from a2-design is presented.

It is known that every self-dual binary code which is not doubly even is a "child" of a doubly even parent. It will be shown that an(n-2,(n-2)/2)child of an(n,n/2,d)doubly even parent has covering radius\geq d-1. Every extremal doubly even(32,16,8)code has covering radius6and every extremal doubly even(48,24,12)code has covering radius8. The complete coset weight distribution of the(32,16,8)quadratic residue code is given, as well as bounds or exact values for the covering radii of all extremai doubly even codes of length less than or equal to96.

The coseta + Aof the linear codeAhas weight distribution related to that ofA^{\perp}and ofA^{\perp} \cap a^{\perp}.

The coset leader of greatest weight in the 3-error-correcting BCH code of length2^{m}-1has weight 5, for oddm \geq 5.