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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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[9]

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

[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.

[11]

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

[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.

[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.

[14]

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

[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.

[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.

[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.

[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.

[19]

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

[20]

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

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[28]

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

[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.

[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.

[31]

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

[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.

[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.

[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.

[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.

[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.

[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.

[38]

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

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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[9]

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

[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.

[11]

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

[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.

[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.

[14]

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

[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.

[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.

[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.

[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.

[19]

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

[20]

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

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[28]

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

[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.

[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.

[31]

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

[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.

[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.

[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.

[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.

[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.

[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.

[38]

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

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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