Maximum Weight Spectrum Codes
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We prove a conjecture recently proposed by Shi, Zhu, Solé and Cohen on linear codes over a finite field F_q. In that work the authors studied the weight set of an [n,k]_q linear code, that is the set of non-zero distinct Hamming weights, showing that its cardinality is upper bounded by q^k-1/q-1. They conjectured that the bound is sharp for every prime power q and every positive integer k , managing to give a proof only when q=2 or k=2. In this work we give two different constructions of codes meeting this bound for every q and k.
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