Bounds on the dimension of codes and subcodes with prescribed contraction index

Let C be a linear code over GF(q), spanned by the rows of a matrix G of rank k. A nonnegative integer λ is said to be the contraction index of C if a maximal set of pairwise linearly independent columns of G has k + λ elements. We derive several upper and lower bounds on the dimension of a proper subcode of C with a prescribed contraction index v < λ. We also present an upper bound on the dimension of any linear code over GF(q) of length n, minimum Hamming distance d, and contraction index λ. For certain values of n and d the latter bound is shown to be tight for all q and λ. This substantially generalizes the results obtained by Delsarte and by Duc for λ = 1.