Small weight codewords of projective geometric codes II
Sam Adriaensen, Lins Denaux

TL;DR
This paper proves that for certain projective geometric codes over composite prime powers, all codewords below a specific weight threshold can be expressed as linear combinations of a limited number of rows, extending previous results.
Contribution
It establishes that for q ≥ 32, all codewords of weight up to O(q^k√q) are linear combinations of at most √q rows, generalizing prior findings to more complex codes.
Findings
Codewords of weight ≤ O(q^k√q) are linear combinations of ≤ √q rows.
Results apply to codes over composite prime powers q ≥ 32.
Generalization to codes involving j-spaces and k-spaces.
Abstract
The -ary linear code is defined as the row space of the incidence matrix of -spaces and points of . It is known that if is square, a codeword of weight exists that cannot be written as a linear combination of at most rows of . 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 is a composite prime power, every codeword of up to weight is a linear combination of at most rows of . We also generalise this result to the codes , which are defined as the -ary row span of the incidence matrix of -spaces and -spaces, .
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Taxonomy
TopicsCoding theory and cryptography · graph theory and CDMA systems · Finite Group Theory Research
