Stability of k mod p multisets and small weight codewords of the code generated by the lines of PG(2,q)

Szőnyi, Tamás; Weiner, Zsuzsa

doi:10.1016/j.jcta.2018.02.005

Cited by 17 publications

(13 citation statements)

References 8 publications

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“…By construction, it is easy to see that this generalised example has some interesting properties. For somewhat larger values of p, Szőnyi and Weiner [13]…”

Section: Results In the Planementioning

confidence: 95%

“…The prime case kept on inspiring more mathematicians. Next in line is Bagchi [3] on the one hand, and Szőnyi and Weiner [13] on the other hand (see Theorem 2.2.5). Bagchi proved the following: Theorem 2.2.1 ([3, Theorem 1.1]).…”

Section: Results In the Planementioning

confidence: 99%

“…As each of the three lines m, m ′ and m ′′ contains p − 1 points with pairwise different, non-zero values, it is easy to see that such a code word can never be written as a linear combination of less than p − 1 different lines. As noted by Szőnyi and Weiner [13], the above example can be generalised as follows: is a code word of weight 3p − 3 or 3p − 2, depending on the value of λ + λ ′ + λ ′′ .…”

Section: Results In the Planementioning

confidence: 99%

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Small weight code words arising from the incidence of points and hyperplanes in $$\text {PG}(n,q)$$

Adriaensen

Denaux

Storme

et al. 2020

Des. Codes Cryptogr.

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Let C n−1 (n, q) be the code arising from the incidence of points and hyperplanes in the Desarguesian projective space PG(n, q). Recently, Polverino and Zullo [12] proved that within this code, all non-zero code words of weight at most 2q n−1 are scalar multiples of either the incidence vector of one hyperplane, or the difference of the incidence vectors of two distinct hyperplanes. We improve this result, proving that when q > 17 and q / ∈ {25, 27, 29, 31, 32, 49, 121}, all code words of weight at most (4q − √ 8q − 33 2 )q n−2 are linear combinations of incidence vectors of hyperplanes through a fixed (n − 3)-space. Depending on the omitted value for q, we can lower the bound on the weight of c to obtain the same results.

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“…By construction, it is easy to see that this generalised example has some interesting properties. For somewhat larger values of p, Szőnyi and Weiner [13]…”

Section: Results In the Planementioning

confidence: 95%

Section: Results In the Planementioning

confidence: 99%

Section: Results In the Planementioning

confidence: 99%

See 1 more Smart Citation

Small weight code words arising from the incidence of points and hyperplanes in $$\text {PG}(n,q)$$

Adriaensen

Denaux

Storme

et al. 2020

Des. Codes Cryptogr.

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“…Let M be a multiset in PG(2, q), 17 < q, so that the number of lines intersecting it in not k (mod p) points is δ, where δ < 3 16 (q + 1) 2 . Then the number of non k (mod p) secants through any point is at 15]). Let M be a multiset in PG(2, q), q = p h , where p is prime.…”

Section: Preliminariesmentioning

confidence: 99%

A Characterization of Hermitian Varieties as Codewords

Aguglia

Bartoli

Storme

et al. 2018

Electron. J. Combin.

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It is known that the Hermitian varieties are codewords in the code defined by the points and hyperplanes of the projective spaces PG(r, q 2 ). In finite geometry, also quasi-Hermitian varieties are defined. These are sets of points of PG(r, q 2 ) of the same size as a non-singular Hermitian variety of PG(r, q 2 ), having the same intersection sizes with the hyperplanes of PG(r, q 2 ). In the planar case, this reduces to the definition of a unital. A famous result of Blokhuis, Brouwer, and Wilbrink states that every unital in the code of the points and lines of PG(2, q 2 ) is a Hermitian curve. We prove a similar result for the quasi-Hermitian varieties in PG(3, q 2 ), q = p h , as well as in PG(r, q 2 ), q = p prime, or q = p 2 , p prime, and r ≥ 4.

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“…During the proof of the main theorem, we will make use of induction on the dimension n. The base case, stated below in Lemma 2.3, was already proved in [25].…”

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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Stability of k mod p multisets and small weight codewords of the code generated by the lines of PG(2,q)

Cited by 17 publications

References 8 publications

Small weight code words arising from the incidence of points and hyperplanes in $$\text {PG}(n,q)$$

Small weight code words arising from the incidence of points and hyperplanes in $$\text {PG}(n,q)$$

A Characterization of Hermitian Varieties as Codewords

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

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