Generalized Pair Weights of Linear Codes and MacWilliams Extension Theorem
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In this paper, we introduce the notion of generalized pair weights of an [n, k]-linear code over a finite field and the notion of pair r-equiweight codes, where 1< r< k-1. We give some properties of generalized pair weights of linear codes over finite fields. We obtain a necessary and sufficient condition for an [n,k]-linear code to be a pair equiweight code and characterize the pair r-equiweight codes for any 1< r< k-1. In addition, a relationship between the pair equiweight code and the pair r-equiweight code is also given. Finally, we prove the MacWilliams extension theorem for the pair weight case.
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