Abstract
Let \(\mathcal{V} \) be a list of all words of \((GF(2))^n \), lexicographically ordered with respect to some basis. Lexicodes are codes constructed from \(\mathcal{V} \) by applying a greedy algorithm. A short proof, only based on simple principles from linear algebra, is given for the linearity of these codes. The proof holds for any ordered basis, and for any selection criterion, thus generalizing the results of several authors. An extension of the applied technique shows that lexicodes over \(GF(2^{2^k } ) \) are linear for a wide choice of bases and for a large class of selection criteria. This result generalizes a property of Conway and Sloane.
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Zanten, A.J.v. Lexicographic Order and Linearity. Designs, Codes and Cryptography 10, 85–97 (1997). https://doi.org/10.1023/A:1008244404559
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DOI: https://doi.org/10.1023/A:1008244404559