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doi: 10.3934/amc.2020129
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Codes with few weights arising from linear sets

Vito Napolitano and  Ferdinando Zullo ,

Dipartimento di Matematica e Fisica, , Università degli Studi della Campania "Luigi Vanvitelli", I– 81100 Caserta, Italy

* Corresponding author: Ferdinando Zullo

Received  May 2020 Revised  October 2020 Early access December 2020

Fund Project: This research was partially supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA - INdAM). The authors were also supported by the project "VALERE: VAnviteLli pEr la RicErca" of the University of Campania "Luigi Vanvitelli"

In this article we present a class of codes with few weights arising from a special type of linear sets. We explicitly show the weights of such codes, their weight enumerators and possible choices for their generator matrices. In particular, our construction yields linear codes with three weights and, in some cases, almost MDS codes. The interest for these codes relies on their applications to authentication codes and secret schemes, and their connections with further objects such as association schemes and graphs.

Keywords: Codes with few weights, linear set, MRD-code, scattered space, projective system.
Mathematics Subject Classification: 51E20, 05B25, 51E22.
Citation: Vito Napolitano, Ferdinando Zullo. Codes with few weights arising from linear sets. Advances in Mathematics of Communications, doi: 10.3934/amc.2020129
References:
[1]

A. Aguglia and L. Giuzzi, Intersection sets, three-character multisets and associated codes, Des. Codes Cryptogr., 83 (2017), 269-282.  doi: 10.1007/s10623-016-0302-8.  Google Scholar

[2]

T. L. Alderson, A note on full weight spectrum codes, Trans. on Combinatorics, 8 (2019), 15-22.  doi: 10.22108/toc.2019.112621.1584.  Google Scholar

[3]

D. Bartoli, C. Zanella and F. Zullo, A new family of maximum scattered linear sets in $\text{PG}(1, q^6)$, Ars Math. Contemp., 19 (2020), 125-145.  doi: 10.26493/1855-3974.2137.7fa.  Google Scholar

[4]

A. Blokhuis and M. Lavrauw, Scattered spaces with respect to a spread in $\text{PG}(n, q)$, Geom. Dedicata, 81 (2000), 231-243.  doi: 10.1023/A:1005283806897.  Google Scholar

[5]

R. Calderbank and J. M. Goethals, Three-weight codes and association schemes, Philips J. Res., 39 (1984), 143-152.   Google Scholar

[6]

A. R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. London Math. Soc., 18 (1986), 97-122.  doi: 10.1112/blms/18.2.97.  Google Scholar

[7]

B. Csajbók, G. Marino, O. Polverino and C. Zanella, A new family of MRD-codes, Linear Algebra Appl., 548 (2018), 203-220.  doi: 10.1016/j.laa.2018.02.027.  Google Scholar

[8]

B. Csajbók, G. Marino, O. Polverino and Y. Zhou, Maximum Rank-Distance codes with maximum left and right idealisers, Discrete Math., 343 (2020), 111985, 16pp. doi: 10.1016/j.disc.2020.111985.  Google Scholar

[9]

B. Csajbók, G. Marino, O. Polverino and F. Zullo, Generalising the scattered property of subspaces, in Combinatorica, arXiv: 1906.10590. Google Scholar

[10]

B. Csajbók, G. Marino and F. Zullo, New maximum scattered linear sets of the projective line, Finite Fields Appl., 54 (2018), 133-150.  doi: 10.1016/j.ffa.2018.08.001.  Google Scholar

[11]

M. A. de Boer, Almost MDS codes, Des. Codes Cryptogr., 9 (1996), 143-155.  doi: 10.1007/BF00124590.  Google Scholar

[12]

P. Delsarte, Bilinear forms over a finite field, with applications to coding theory, J. Combin. Theory Ser. A, 25 (1978), 226-241.  doi: 10.1016/0097-3165(78)90015-8.  Google Scholar

[13]

K. Ding and C. Ding, A class of two-weight and three-weight codes and their applications in secret sharing, IEEE Trans. Inform. Theory, 61 (2015), 5835-5842.  doi: 10.1109/TIT.2015.2473861.  Google Scholar

[14]

K. Ding K. and C. Ding, Binary linear codes with three weights, IEEE Commun. Lett., 18 (2014), 1879-1882.   Google Scholar

[15]

C. Ding, C. Li, N. Li and Z. Zhou, Three-weight cyclic codes and their weight distributions, Discret. Math., 339 (2016), 415-427.  doi: 10.1016/j.disc.2015.09.001.  Google Scholar

[16]

C. Ding, J. Luo and H. Niederreiter, Two-weight codes punctured from irreducible cyclic codes, Proc. Ist Int. Workshop Coding theory and Cryptogr., (2008), 119–124. doi: 10.1142/9789812832245_0009.  Google Scholar

[17]

C. Ding and H. Niederreiter, Cyclotomic linear codes of order $3$, IEEE Trans. Inf. Theory, 53 (2007), 2274-2277.  doi: 10.1109/TIT.2007.896886.  Google Scholar

[18]

C. Ding and X. Wang, A coding theory construction of new systematic authentication codes, Theoretical computer science, 330 (2005), 81-99.  doi: 10.1016/j.tcs.2004.09.011.  Google Scholar

[19]

N. Durante, On sets with few intersection numbers in finite projective and affine spaces, Electron. J. Combin., 21 (2014), 4.13, 18 pp.  Google Scholar

[20]

È. Gabidulin, Theory of codes with maximum rank distance, Problems of Information Transmission, 21 (1985), 3-16.   Google Scholar

[21]

A. Kshevetskiy and E. Gabidulin, The new construction of rank codes, International Symposium on Information Theory, (2005), 2105–2108. doi: 10.1109/ISIT.2005.1523717.  Google Scholar

[22]

M. Lavrauw, Scattered Spaces with Respect to Spreads, and Eggs in Finite Projective Spaces, Ph.D thesis, Eindhoven University of Technology, 2001.  Google Scholar

[23]

D. Liebhold and G. Nebe, Automorphism groups of Gabidulin-like codes, Arch. Math., 107 (2016), 355-366.  doi: 10.1007/s00013-016-0949-4.  Google Scholar

[24]

G. Lunardon, MRD-codes and linear sets, J. Combin. Theory Ser. A, 149 (2017), 1-20.  doi: 10.1016/j.jcta.2017.01.002.  Google Scholar

[25]

G. Lunardon, R. Trombetti and Y. Zhou, Generalized twisted gabidulin codes, J. Combin. Theory Ser. A, 159 (2018), 79-106.  doi: 10.1016/j.jcta.2018.05.004.  Google Scholar

[26]

G. Lunardon, R. Trombetti and Y. Zhou, On kernels and nuclei of rank metric codes, J. Algebraic Combin., 46 (2017), 313-340.  doi: 10.1007/s10801-017-0755-5.  Google Scholar

[27]

S. Mehta, V. Saraswat and S. Sen, Secret sharing using near-MDS codes, Codes, Cryptology, and Information Security (C2SI 2019), LNCS, Springer, 11445 (2019), 195–214.  Google Scholar

[28]

V. Napolitano, O. Polverino, G. Zini and F. Zullo, Linear sets from projection of Desarguesian spreads, arXiv: 2001.08685. Google Scholar

[29]

G. Marino, M. Montanucci and F. Zullo, MRD-codes arising from the trinomial $x^q + x^{q^3}+ cx^{q^5} \in {\mathbb F}_{q^6}[x]$, Linear Algebra Appl., 591 (2020), 99-114.  doi: 10.1016/j.laa.2020.01.004.  Google Scholar

[30]

O. Polverino and F. Zullo, On the number of roots of some linearized polynomials, Linear Algebra Appl., 601 (2020), 189-218.  doi: 10.1016/j.laa.2020.05.009.  Google Scholar

[31]

J. Sheekey, A new family of linear maximum rank distance codes, Adv. Math. Commun., 10 (2016), 475-488.  doi: 10.3934/amc.2016019.  Google Scholar

[32]

J. Sheekey and G. Van de Voorde, Rank-metric codes, linear sets and their duality, Des. Codes Cryptogr., 88 (2020), 655-675.  doi: 10.1007/s10623-019-00703-z.  Google Scholar

[33]

M. Shi and P. Solé, Three-weight codes, triple sum sets, and strongly walk regular graphs, Designs, Codes and Cryptogr., 87 (2019), 2395-2404.  doi: 10.1007/s10623-019-00628-7.  Google Scholar

[34]

M. Tsfasman, S. Vlăduţ and D. Nogin, Algebraic Geometric Codes: Basic Notions, Mathematical Surveys and Monographs, American Mathematical Society, 2007. doi: 10.1090/surv/139.  Google Scholar

[35]

B. Wu and Z. Liu, Linearized polynomials over finite fields revisited, Finite Fields Appl., 22 (2013), 79-100.  doi: 10.1016/j.ffa.2013.03.003.  Google Scholar

[36]

Y. Wu, Q. Yansheng and X. Shi, At most three-weight binary linear codes from generalized Moisio's exponential sums, Designs, Codes and Cryptogr., 87 (2019), 1927-1943.  doi: 10.1007/s10623-018-00595-5.  Google Scholar

[37]

C. Zanella and F. Zullo, Vertex properties of maximum scattered linear sets of $\text{PG}(1, q^n)$, Discrete Math., 343 (2020), 111800, 14pp. doi: 10.1016/j.disc.2019.111800.  Google Scholar

[38]

G. Zini and F. Zullo, Scattered subspaces and related codes, arXiv: 2007.04643. Google Scholar

show all references

References:
[1]

A. Aguglia and L. Giuzzi, Intersection sets, three-character multisets and associated codes, Des. Codes Cryptogr., 83 (2017), 269-282.  doi: 10.1007/s10623-016-0302-8.  Google Scholar

[2]

T. L. Alderson, A note on full weight spectrum codes, Trans. on Combinatorics, 8 (2019), 15-22.  doi: 10.22108/toc.2019.112621.1584.  Google Scholar

[3]

D. Bartoli, C. Zanella and F. Zullo, A new family of maximum scattered linear sets in $\text{PG}(1, q^6)$, Ars Math. Contemp., 19 (2020), 125-145.  doi: 10.26493/1855-3974.2137.7fa.  Google Scholar

[4]

A. Blokhuis and M. Lavrauw, Scattered spaces with respect to a spread in $\text{PG}(n, q)$, Geom. Dedicata, 81 (2000), 231-243.  doi: 10.1023/A:1005283806897.  Google Scholar

[5]

R. Calderbank and J. M. Goethals, Three-weight codes and association schemes, Philips J. Res., 39 (1984), 143-152.   Google Scholar

[6]

A. R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. London Math. Soc., 18 (1986), 97-122.  doi: 10.1112/blms/18.2.97.  Google Scholar

[7]

B. Csajbók, G. Marino, O. Polverino and C. Zanella, A new family of MRD-codes, Linear Algebra Appl., 548 (2018), 203-220.  doi: 10.1016/j.laa.2018.02.027.  Google Scholar

[8]

B. Csajbók, G. Marino, O. Polverino and Y. Zhou, Maximum Rank-Distance codes with maximum left and right idealisers, Discrete Math., 343 (2020), 111985, 16pp. doi: 10.1016/j.disc.2020.111985.  Google Scholar

[9]

B. Csajbók, G. Marino, O. Polverino and F. Zullo, Generalising the scattered property of subspaces, in Combinatorica, arXiv: 1906.10590. Google Scholar

[10]

B. Csajbók, G. Marino and F. Zullo, New maximum scattered linear sets of the projective line, Finite Fields Appl., 54 (2018), 133-150.  doi: 10.1016/j.ffa.2018.08.001.  Google Scholar

[11]

M. A. de Boer, Almost MDS codes, Des. Codes Cryptogr., 9 (1996), 143-155.  doi: 10.1007/BF00124590.  Google Scholar

[12]

P. Delsarte, Bilinear forms over a finite field, with applications to coding theory, J. Combin. Theory Ser. A, 25 (1978), 226-241.  doi: 10.1016/0097-3165(78)90015-8.  Google Scholar

[13]

K. Ding and C. Ding, A class of two-weight and three-weight codes and their applications in secret sharing, IEEE Trans. Inform. Theory, 61 (2015), 5835-5842.  doi: 10.1109/TIT.2015.2473861.  Google Scholar

[14]

K. Ding K. and C. Ding, Binary linear codes with three weights, IEEE Commun. Lett., 18 (2014), 1879-1882.   Google Scholar

[15]

C. Ding, C. Li, N. Li and Z. Zhou, Three-weight cyclic codes and their weight distributions, Discret. Math., 339 (2016), 415-427.  doi: 10.1016/j.disc.2015.09.001.  Google Scholar

[16]

C. Ding, J. Luo and H. Niederreiter, Two-weight codes punctured from irreducible cyclic codes, Proc. Ist Int. Workshop Coding theory and Cryptogr., (2008), 119–124. doi: 10.1142/9789812832245_0009.  Google Scholar

[17]

C. Ding and H. Niederreiter, Cyclotomic linear codes of order $3$, IEEE Trans. Inf. Theory, 53 (2007), 2274-2277.  doi: 10.1109/TIT.2007.896886.  Google Scholar

[18]

C. Ding and X. Wang, A coding theory construction of new systematic authentication codes, Theoretical computer science, 330 (2005), 81-99.  doi: 10.1016/j.tcs.2004.09.011.  Google Scholar

[19]

N. Durante, On sets with few intersection numbers in finite projective and affine spaces, Electron. J. Combin., 21 (2014), 4.13, 18 pp.  Google Scholar

[20]

È. Gabidulin, Theory of codes with maximum rank distance, Problems of Information Transmission, 21 (1985), 3-16.   Google Scholar

[21]

A. Kshevetskiy and E. Gabidulin, The new construction of rank codes, International Symposium on Information Theory, (2005), 2105–2108. doi: 10.1109/ISIT.2005.1523717.  Google Scholar

[22]

M. Lavrauw, Scattered Spaces with Respect to Spreads, and Eggs in Finite Projective Spaces, Ph.D thesis, Eindhoven University of Technology, 2001.  Google Scholar

[23]

D. Liebhold and G. Nebe, Automorphism groups of Gabidulin-like codes, Arch. Math., 107 (2016), 355-366.  doi: 10.1007/s00013-016-0949-4.  Google Scholar

[24]

G. Lunardon, MRD-codes and linear sets, J. Combin. Theory Ser. A, 149 (2017), 1-20.  doi: 10.1016/j.jcta.2017.01.002.  Google Scholar

[25]

G. Lunardon, R. Trombetti and Y. Zhou, Generalized twisted gabidulin codes, J. Combin. Theory Ser. A, 159 (2018), 79-106.  doi: 10.1016/j.jcta.2018.05.004.  Google Scholar

[26]

G. Lunardon, R. Trombetti and Y. Zhou, On kernels and nuclei of rank metric codes, J. Algebraic Combin., 46 (2017), 313-340.  doi: 10.1007/s10801-017-0755-5.  Google Scholar

[27]

S. Mehta, V. Saraswat and S. Sen, Secret sharing using near-MDS codes, Codes, Cryptology, and Information Security (C2SI 2019), LNCS, Springer, 11445 (2019), 195–214.  Google Scholar

[28]

V. Napolitano, O. Polverino, G. Zini and F. Zullo, Linear sets from projection of Desarguesian spreads, arXiv: 2001.08685. Google Scholar

[29]

G. Marino, M. Montanucci and F. Zullo, MRD-codes arising from the trinomial $x^q + x^{q^3}+ cx^{q^5} \in {\mathbb F}_{q^6}[x]$, Linear Algebra Appl., 591 (2020), 99-114.  doi: 10.1016/j.laa.2020.01.004.  Google Scholar

[30]

O. Polverino and F. Zullo, On the number of roots of some linearized polynomials, Linear Algebra Appl., 601 (2020), 189-218.  doi: 10.1016/j.laa.2020.05.009.  Google Scholar

[31]

J. Sheekey, A new family of linear maximum rank distance codes, Adv. Math. Commun., 10 (2016), 475-488.  doi: 10.3934/amc.2016019.  Google Scholar

[32]

J. Sheekey and G. Van de Voorde, Rank-metric codes, linear sets and their duality, Des. Codes Cryptogr., 88 (2020), 655-675.  doi: 10.1007/s10623-019-00703-z.  Google Scholar

[33]

M. Shi and P. Solé, Three-weight codes, triple sum sets, and strongly walk regular graphs, Designs, Codes and Cryptogr., 87 (2019), 2395-2404.  doi: 10.1007/s10623-019-00628-7.  Google Scholar

[34]

M. Tsfasman, S. Vlăduţ and D. Nogin, Algebraic Geometric Codes: Basic Notions, Mathematical Surveys and Monographs, American Mathematical Society, 2007. doi: 10.1090/surv/139.  Google Scholar

[35]

B. Wu and Z. Liu, Linearized polynomials over finite fields revisited, Finite Fields Appl., 22 (2013), 79-100.  doi: 10.1016/j.ffa.2013.03.003.  Google Scholar

[36]

Y. Wu, Q. Yansheng and X. Shi, At most three-weight binary linear codes from generalized Moisio's exponential sums, Designs, Codes and Cryptogr., 87 (2019), 1927-1943.  doi: 10.1007/s10623-018-00595-5.  Google Scholar

[37]

C. Zanella and F. Zullo, Vertex properties of maximum scattered linear sets of $\text{PG}(1, q^n)$, Discrete Math., 343 (2020), 111800, 14pp. doi: 10.1016/j.disc.2019.111800.  Google Scholar

[38]

G. Zini and F. Zullo, Scattered subspaces and related codes, arXiv: 2007.04643. Google Scholar

Table 1.  Possible choices for $ f_1, \ldots, f_r $
$ n $ $ r $ $ (f_1(x), \ldots, f_r(x)) $ conditions references
$ (x, x^{q^s}, \ldots, x^{q^{s(r-1)}}) $ $ \gcd(s, n)=1 $ [12,20,21]
$ (x^{q^s}, \ldots, x^{q^{s(r-2)}}, x+\delta x^{q^{s(r-1)}}) $ $ \begin{array}{cc} \gcd(s, n)=1, \\ \mathrm{N}_{q^n/q}(\delta)\neq (-1)^{nr}\end{array} $ [31,25]
$ 6 $ $ 4 $ $ (x^q, x^{q^2}, x^{q^4}, x-\delta^{q^5} x^{q^{3}}) $ $ q >4\\ \text{certain}\ \ \text{ choices}\ \text{ of} \, \delta $ [7,30]
$ 6 $ $ 4 $ $ (x^q, x^{q^3}, x-x^{q^2}, x^{q^4}-\delta x) $ $ \begin{array}{cccc}q \quad \text{odd}\\ \delta^2+\delta =1 \end{array} $ [10,29]
$ 6 $ $ 4 $ $ \begin{array}{cc} (h^{q^2-1}x^q+h^{q-1}x^{q^2}, x^{q^3}, \\ x^q-h^{q-1}x^{q^4}, x^q-h^{q-1}x^{q^5}) \end{array} $ $ \begin{array}{cccc}q \quad \text{odd}\\ h^{q^3+1}=-1 \end{array} $ [3,37]
$ 7 $ $ 3 $ $ (x, x^q, x^{q^3}) $ $ \begin{array}{cc} q \quad \text{odd}, \\ \gcd(s, 7)=1\end{array} $ [8]
$ 7 $ $ 4 $ $ (x, x^{q^{2s}}, x^{q^{3s}}, x^{q^{4s}}) $ $ \begin{array}{cc} q \quad \text{odd}, \\ \gcd(s, n)=1\end{array} $ [8]
$ 8 $ $ 3 $ $ (x, x^q, x^{q^3}) $ $ \begin{array}{cc} q \equiv 1 \pmod{3}, \\ \gcd(s, 8)=1\end{array} $ [8]
$ 8 $ $ 5 $ $ (x, x^{q^{2s}}, x^{q^{3s}}, x^{q^{4s}}, x^{{5s}}) $ $ \begin{array}{cc} q \equiv 1 \pmod{3}, \\ \gcd(s, 8)=1 \end{array} $ [8]
$ 8 $ $ 6 $ $ (x^q, x^{q^2}, x^{q^3}, x^{q^5}, x^{q^6}, x-\delta x^{q^4}) $ $ \begin{array}{cc} q\, \text{odd}, \\ \delta^2=-1\end{array} $ [7]
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$ n $ $ r $ $ (f_1(x), \ldots, f_r(x)) $ conditions references
$ (x, x^{q^s}, \ldots, x^{q^{s(r-1)}}) $ $ \gcd(s, n)=1 $ [12,20,21]
$ (x^{q^s}, \ldots, x^{q^{s(r-2)}}, x+\delta x^{q^{s(r-1)}}) $ $ \begin{array}{cc} \gcd(s, n)=1, \\ \mathrm{N}_{q^n/q}(\delta)\neq (-1)^{nr}\end{array} $ [31,25]
$ 6 $ $ 4 $ $ (x^q, x^{q^2}, x^{q^4}, x-\delta^{q^5} x^{q^{3}}) $ $ q >4\\ \text{certain}\ \ \text{ choices}\ \text{ of} \, \delta $ [7,30]
$ 6 $ $ 4 $ $ (x^q, x^{q^3}, x-x^{q^2}, x^{q^4}-\delta x) $ $ \begin{array}{cccc}q \quad \text{odd}\\ \delta^2+\delta =1 \end{array} $ [10,29]
$ 6 $ $ 4 $ $ \begin{array}{cc} (h^{q^2-1}x^q+h^{q-1}x^{q^2}, x^{q^3}, \\ x^q-h^{q-1}x^{q^4}, x^q-h^{q-1}x^{q^5}) \end{array} $ $ \begin{array}{cccc}q \quad \text{odd}\\ h^{q^3+1}=-1 \end{array} $ [3,37]
$ 7 $ $ 3 $ $ (x, x^q, x^{q^3}) $ $ \begin{array}{cc} q \quad \text{odd}, \\ \gcd(s, 7)=1\end{array} $ [8]
$ 7 $ $ 4 $ $ (x, x^{q^{2s}}, x^{q^{3s}}, x^{q^{4s}}) $ $ \begin{array}{cc} q \quad \text{odd}, \\ \gcd(s, n)=1\end{array} $ [8]
$ 8 $ $ 3 $ $ (x, x^q, x^{q^3}) $ $ \begin{array}{cc} q \equiv 1 \pmod{3}, \\ \gcd(s, 8)=1\end{array} $ [8]
$ 8 $ $ 5 $ $ (x, x^{q^{2s}}, x^{q^{3s}}, x^{q^{4s}}, x^{{5s}}) $ $ \begin{array}{cc} q \equiv 1 \pmod{3}, \\ \gcd(s, 8)=1 \end{array} $ [8]
$ 8 $ $ 6 $ $ (x^q, x^{q^2}, x^{q^3}, x^{q^5}, x^{q^6}, x-\delta x^{q^4}) $ $ \begin{array}{cc} q\, \text{odd}, \\ \delta^2=-1\end{array} $ [7]
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