Small weight codewords of projective geometric codes II

Sam Adriaensen, Lins Denaux

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Abstract

The p-ary linear code Ckn,q is defined as the row space of the incidence matrix A of k-spaces and points of PGn,q. It is known that if q is square, a codeword of weight qkq+Oqk-1 exists that cannot be written as a linear combination of at most q rows of A. Over the past few decades, researchers have put a lot of effort towards proving that any codeword of smaller weight does meet this property. We show that if q⩾32 is a composite prime power, every codeword of Ckn,q up to weight Oqkq is a linear combination of at most q rows of A. We also generalise this result to the codes Cj,kn,q, which are defined as the p-ary row span of the incidence matrix of k-spaces and j-spaces, j<k.

Original languageEnglish
Pages (from-to)2451-2472
Number of pages22
JournalDesigns, Codes and Cryptography
Volume92
Issue number9
DOIs
Publication statusPublished - 28 Apr 2024

Bibliographical note

Publisher Copyright:
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024.

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