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

Alexander Vardy, Jakov Snyders, Yair Be'ery
1990 Linear Algebra and its Applications  
Let % be a linear code over GF(q), spanned by the rows of a matrix G of rank k. A nonnegative integer A is said to be the contraction index of 8 if a maximal set of pairwise linearly independent columns of G has k + A elements. We derive several upper and lower bounds on the dimension of a proper subcode of 4 with a prescribed contraction index v < A. We also present an upper bound on the dimension of any linear code over GF(9) of length n, minimum Hamming distance d, and contraction index A.
more » ... r certain values of n and d the latter bound is shown to be tight for all 9 and A. This substantially generalizes the results obtained by Delsarte andbyDucforA=l. 1. Given a set of nonzero vectors with entries from GF(q), we say that these vectors are pairuhe linearly independent over GF(q) if no vector in the set is a scalar multiple of some other vector in the set. The contraction index A of the code d is defined as h=msax[card(S)-k], LZNEAR ALGEBRA AND ITS APPLZCATZONS 142:237-261(1990) 0
doi:10.1016/0024-3795(90)90269-i fatcat:xmowppho25azbiodnxvv24zk4e