Lecture 6a (draft; 9/21/03). Linear codes. Weights, supports, ranks.

A shortening of C on the coordinate i is the [n − 1, k − 1, d] linear code obtained by leaving only such codewords and deleting the ith coordinate. Let [n] := {1, . . . , n}, E ⊂ [n], Ē = [n]\E. For a matrix M with n columns we denote by M(E) the submatrix formed by all the columns with numbers in E. The support of a vector x ∈ H n q is defined as supp(x) = {i : xi 6= 0}. Given a subcode A ⊂ C, its support is supp(A) = {i ∈ [n] : ∃a∈A ai 6= 0} A shortening of C on the coordinates in Ē is a linear code C such that ∀x∈CE supp(x) ⊂ E. A projection of C on the coordinates in Ē, also called puncturing, is a restriction of C to the coordinates in E (simply said, deleting the coorinates in Ē from every codevector of C and deleting repeated entries from the result). The basic facts about shortenings and puncturings are given in the following