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Small weight codewords of projective geometric codes II

Sam Adriaensen and Lins Denaux
(2024) DESIGNS CODES AND CRYPTOGRAPHY. 92(9). p.2451-2472
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Abstract
The p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document}-ary linear code Ckn,q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}_{k}\!\left( n,q\right) $$\end{document} is defined as the row space of the incidence matrix A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A$$\end{document} of k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-spaces and points of PGn,q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{PG}\!\left( n,q\right) $$\end{document}. It is known that if q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document} is square, a codeword of weight qkq+Oqk-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q<^>k\sqrt{q}+\mathcal {O}\!\left( q<^>{k-1}\right) $$\end{document} exists that cannot be written as a linear combination of at most q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{q}$$\end{document} rows of A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A$$\end{document}. 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\geqslant 32$$\end{document} is a composite prime power, every codeword of Ckn,q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}_{k}\!\left( n,q\right) $$\end{document} up to weight Oqkq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}\!\left( q<^>k\sqrt{q}\right) $$\end{document} is a linear combination of at most q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{q}$$\end{document} rows of A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A$$\end{document}. We also generalise this result to the codes Cj,kn,q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}_{j,k}\!\left( n,q\right) $$\end{document}, which are defined as the p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document}-ary row span of the incidence matrix of k-spaces and j-spaces, j
Keywords
Linear codes, Incidence matrices, Projective spaces, Small weight codewords, INCIDENCE MATRIX, K-SPACES, POINTS, HYPERPLANES, PG(N

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MLA
Adriaensen, Sam, and Lins Denaux. “Small Weight Codewords of Projective Geometric Codes II.” DESIGNS CODES AND CRYPTOGRAPHY, vol. 92, no. 9, 2024, pp. 2451–72, doi:10.1007/s10623-024-01397-8.
APA
Adriaensen, S., & Denaux, L. (2024). Small weight codewords of projective geometric codes II. DESIGNS CODES AND CRYPTOGRAPHY, 92(9), 2451–2472. https://doi.org/10.1007/s10623-024-01397-8
Chicago author-date
Adriaensen, Sam, and Lins Denaux. 2024. “Small Weight Codewords of Projective Geometric Codes II.” DESIGNS CODES AND CRYPTOGRAPHY 92 (9): 2451–72. https://doi.org/10.1007/s10623-024-01397-8.
Chicago author-date (all authors)
Adriaensen, Sam, and Lins Denaux. 2024. “Small Weight Codewords of Projective Geometric Codes II.” DESIGNS CODES AND CRYPTOGRAPHY 92 (9): 2451–2472. doi:10.1007/s10623-024-01397-8.
Vancouver
1.
Adriaensen S, Denaux L. Small weight codewords of projective geometric codes II. DESIGNS CODES AND CRYPTOGRAPHY. 2024;92(9):2451–72.
IEEE
[1]
S. Adriaensen and L. Denaux, “Small weight codewords of projective geometric codes II,” DESIGNS CODES AND CRYPTOGRAPHY, vol. 92, no. 9, pp. 2451–2472, 2024.
@article{01HWNEC3RKXXXJ6ZQPJY0JSR3P,
  abstract     = {{The p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document}-ary linear code Ckn,q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}_{k}\!\left( n,q\right) $$\end{document} is defined as the row space of the incidence matrix A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A$$\end{document} of k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-spaces and points of PGn,q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{PG}\!\left( n,q\right) $$\end{document}. It is known that if q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document} is square, a codeword of weight qkq+Oqk-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q<^>k\sqrt{q}+\mathcal {O}\!\left( q<^>{k-1}\right) $$\end{document} exists that cannot be written as a linear combination of at most q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{q}$$\end{document} rows of A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A$$\end{document}. 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\geqslant 32$$\end{document} is a composite prime power, every codeword of Ckn,q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}_{k}\!\left( n,q\right) $$\end{document} up to weight Oqkq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}\!\left( q<^>k\sqrt{q}\right) $$\end{document} is a linear combination of at most q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{q}$$\end{document} rows of A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A$$\end{document}. We also generalise this result to the codes Cj,kn,q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}_{j,k}\!\left( n,q\right) $$\end{document}, which are defined as the p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document}-ary row span of the incidence matrix of k-spaces and j-spaces, j}},
  author       = {{Adriaensen, Sam and Denaux, Lins}},
  issn         = {{0925-1022}},
  journal      = {{DESIGNS CODES AND CRYPTOGRAPHY}},
  keywords     = {{Linear codes,Incidence matrices,Projective spaces,Small weight codewords,INCIDENCE MATRIX,K-SPACES,POINTS,HYPERPLANES,PG(N}},
  language     = {{eng}},
  number       = {{9}},
  pages        = {{2451--2472}},
  title        = {{Small weight codewords of projective geometric codes II}},
  url          = {{http://doi.org/10.1007/s10623-024-01397-8}},
  volume       = {{92}},
  year         = {{2024}},
}
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