Small weight codewords of projective geometric codes

Adriaensen, Sam; Denaux, Lins

doi:10.1016/j.jcta.2020.105395

Cited by 6 publications

(7 citation statements)

References 24 publications

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“…In general, finding the minimum distance of a linear code and classifying its nonzero codewords of minimum weight is not an easy task. Even for linear codes from geometric constructions, it is often highly non-trivial to find sharp bounds or a classification of the smallest weight words, see for example [1,3,17,26,35].…”

Section: Coding Theory and Moderate-density Parity-check Codesmentioning

confidence: 99%

“…The family of MDPC codes that we are considering is built upon a parity-check matrix as in (1). In such a matrix the number of columns is twice the number of rows and this coincides with the setting originally studied in [24].…”

Section: Remark 49mentioning

confidence: 99%

“…We would like to focus in this subsection here on the performance of the constructed MDPC code C 2 (Π Γ ) ⊥ within one round of the bit-flipping decoding algorithm. We now adapt and apply Theorem 2.4 to the parity-check matrix H of C 2 (Π Γ ) ⊥ given in (1).…”

Section: Error-correction Capabilitymentioning

confidence: 99%

“…Therefore, a second round of the bit-flipping is able to correct completely if after one round there are no more than q+1 4 errors left. Applying (6) for t = q+1 4 to the parity-check matrix H of C 2 (Π Γ ) ⊥ given in (1), we obtain that we can successfully correct every error of weight t = ( √ n) after two rounds of the bit-flipping decoding algorithm with probability e − (n) .…”

Section: Remark 416mentioning

confidence: 99%

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Moderate-density parity-check codes from projective bundles

Jessica

Mattheus

Neri

et al. 2022

Des. Codes Cryptogr.

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New constructions for moderate-density parity-check (MDPC) codes using finite geometry are proposed. We design a parity-check matrix for the main family of binary codes as the concatenation of two matrices: the incidence matrix between points and lines of the Desarguesian projective plane and the incidence matrix between points and ovals of a projective bundle. A projective bundle is a special collection of ovals which pairwise meet in a unique point. We determine the minimum distance and the dimension of these codes, and we show that they have a natural quasi-cyclic structure. We consider alternative constructions based on an incidence matrix of a Desarguesian projective plane and compare their error-correction performance with regards to a modification of Gallager’s bit-flipping decoding algorithm. In this setting, our codes have the best possible error-correction performance after one round of bit-flipping decoding given the parameters of the code’s parity-check matrix.

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Section: Coding Theory and Moderate-density Parity-check Codesmentioning

confidence: 99%

Section: Remark 49mentioning

confidence: 99%

Section: Error-correction Capabilitymentioning

confidence: 99%

Section: Remark 416mentioning

confidence: 99%

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Moderate-density parity-check codes from projective bundles

Jessica

Mattheus

Neri

et al. 2022

Des. Codes Cryptogr.

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“…Note that if q ∈ {16, 27} and n = 3, then Theorem 1.5 follows from the (j, k) = (0, 2) case of Result 1.1; all other values of q and n 3 not satisfying (1) provide non-positive upper bounds on the weights wt(c) and hence make Theorem 1.5 trivially true. This means that the explicit assumptions on q isn't necessary for the theorem to stay true; we however keep the assumptions (1) to emphasize that we may assume q to be big.…”

Section: Introductionmentioning

confidence: 99%

Minimal codewords arising from the incidence of points and hyperplanes in projective spaces

Bartoli¹,

Denaux²

2023

AMC

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<p style='text-indent:20px;'>Over the past few years, the codes <inline-formula><tex-math id="M1">\begin{document}$ {\mathcal{C}}_{n-1}(n,q) $\end{document}</tex-math></inline-formula> arising from the incidence of points and hyperplanes in the projective space <inline-formula><tex-math id="M2">\begin{document}$ {\rm{PG}}(n,q) $\end{document}</tex-math></inline-formula> attracted a lot of attention. In particular, small weight codewords of <inline-formula><tex-math id="M3">\begin{document}$ {\mathcal{C}}_{n-1}(n,q) $\end{document}</tex-math></inline-formula> are a topic of investigation. The main result of this work states that, if <inline-formula><tex-math id="M4">\begin{document}$ q $\end{document}</tex-math></inline-formula> is large enough and not prime, a codeword having weight smaller than roughly <inline-formula><tex-math id="M5">\begin{document}$ \frac{1}{2^{n-2}}q^{n-1}\sqrt{q} $\end{document}</tex-math></inline-formula> can be written as a linear combination of a few hyperplanes. Consequently, we use this result to provide a graph-theoretical sufficient condition for these codewords of small weight to be minimal.</p>

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