doi: 10.3934/amc.2022020
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Three weight ternary linear codes from non-weakly regular bent functions

Erzurum Technical University, Yakutiye, Erzurum, Turkey

* Corresponding author: Rumi Melih Pelen

Received  May 2021 Revised  December 2021 Early access March 2022

This paper constructs several classes of three-weight ternary linear codes from non-weakly regular dual-bent functions based on a generic construction method. Instead of the whole space, we use the subspaces $ B_{\pm}(f) $ associated with a ternary non-weakly regular dual-bent function $ f $. Unusually, we use the pre-image sets of the dual function $ f^* $ in $ B_{\pm}(f) $ as the defining sets of the corresponding codes. Since the size of the defining sets of the constructed codes is flexible, it enables us to construct several codes with different parameters for a fixed dimension. We represent the weight distribution of the constructed codes, and we also give several examples.

Citation: Rumi Melih Pelen. Three weight ternary linear codes from non-weakly regular bent functions. Advances in Mathematics of Communications, doi: 10.3934/amc.2022020
References:
[1]

R. AndersonC. DingT. Helleseth and T. Klove, How to build robust shared control systems, Designs, Codes and Cryptography, 15 (1998), 111-124.  doi: 10.1023/A:1026421315292.

[2]

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

[3]

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.

[4] C. Carlet, Boolean Functions for Cryptography and Coding Theory, Cambridge University Press, Cambridge, U.K., 2021.  doi: 10.1017/9781108606806.
[5]

C. CarletC. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Transactions on Information Theory, 51 (2005), 2089-2102.  doi: 10.1109/TIT.2005.847722.

[6]

A. Çeşmelioǧlu and W. Meidl, Bent functions of maximal degree, IEEE Transactions on Information Theory, 58 (2012), 1186-1190.  doi: 10.1109/TIT.2011.2170053.

[7]

A. Çeşmelioǧlu and W. Meidl, A construction of bent functions from plateaued functions, Designs, Codes and Cryptography, 66 (2013), 231-242.  doi: 10.1007/s10623-012-9686-2.

[8]

A. ÇeşmelioǧluW. Meidl and A. Pott, Generalized Maiorana-McFarland class and normality of $p$-ary bent functions, Finite Fields and Their Applications, 24 (2013), 105-117.  doi: 10.1016/j.ffa.2013.06.001.

[9]

A. ÇeşmelioǧluW. Meidl and A. Pott, On the dual of (non)-weakly regular bent functions and self-dual bent functions, Advances in Mathematics of Communications, 7 (2013), 425-440.  doi: 10.3934/amc.2013.7.425.

[10]

C. Ding and H. Niederreiter, Cyclotomic linear codes of order 3, IEEE Transactions on Information Theory, 53 (2007), 2274-2277.  doi: 10.1109/TIT.2007.896886.

[11]

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

[12]

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

[13]

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 3rd ed, Oxford, at the Clarendon Press, 1954.

[14]

T. Helleseth and A. Kholosha, Monomial and quadratic bent functions over the finite fields of odd characteristic, IEEE Transactions on Information Theory, 52 (2006), 2018-2032.  doi: 10.1109/TIT.2006.872854.

[15]

T. Helleseth and A. Kholosha, Crosscorrelation of m-sequences exponential sums bent functions and jacobsthal sums, Cryptography and Communications, 3 (2011), 281-291.  doi: 10.1007/s12095-011-0048-0.

[16]

T. Helleseth and A. Kholosha, New binomial bent functions over the finite fields of odd characteristic, IEEE Transactions on Information Theory, 56 (2010), 4646-4652.  doi: 10.1109/TIT.2010.2055130.

[17]

J. Y. HyunJ. Lee and Y. Lee, Explicit criterions for construction of plateaued functions, IEEE Transactions on Information Theory, 62 (2016), 7555-7565.  doi: 10.1109/TIT.2016.2582217.

[18]

I. M. Isaacs, Character Theory of Finite Groups, AMS Chelsea Publishing, Providence, RI, 2006. doi: 10.1090/chel/359.

[19]

P. V. KumarR. A. Scholtz and L. R. Welch, Generalized bent functions and their properties, J. Combinatorial Theory Ser. A, 40 (1985), 90-107.  doi: 10.1016/0097-3165(85)90049-4.

[20]

N. Li and S. Mesnager, Recent results and problems on constructions of linear codes from cryptographic functions, Cryptogr. Commun., 12 (2020), 965-986.  doi: 10.1007/s12095-020-00435-1.

[21]

S. Mesnager, Linear codes with few weights from weakly regular bent functions based on a generic construction, Cryptography and Communications, 9 (2017), 71-84.  doi: 10.1007/s12095-016-0186-5.

[22]

S. Mesnager, Bent Functions: Fundamentals and Results, Springer, [Cham], 2016. doi: 10.1007/978-3-319-32595-8.

[23]

S. Mesnager, Linear Codes from Functions, Chapter 20 in A Concise Encyclopedia of Coding Theory CRC Press/Taylor and Francis Group (Publisher) London, New York, 2021, 94 pp. doi: 10.1201/9781315147901.

[24]

S. MesnagerF. Özbudak and A. Sınak, Linear codes from weakly regular plateaued functions and their secret sharing schemes, Designs, Codes and Cryptography, 87 (2019), 463-480.  doi: 10.1007/s10623-018-0556-4.

[25]

F. Özbudak and R. M. Pelen, Duals of non-weakly regular bent functions are not weakly regular and generalization to plateaued functions, Finite Fields and Their Applications, 64 (2020), 101668, 16 pp. doi: 10.1016/j.ffa.2020.101668.

[26]

F. Özbudak and R. M. Pelen, Strongly regular graphs arising from non-weakly regular bent functions, Cryptography and Communications, 11 (2019), 1297-1306.  doi: 10.1007/s12095-019-00394-2.

[27]

R. M. Pelen, Studies on Non-Weakly Regular Bent Functions and Related Structures, Thesis (Ph.D.) – Graduate School of Natural and Applied Sciences. Mathematics., Middle East Technical University, 2020.

[28]

C. TangN. LiY. QiZ. Zhou and and T. Helleseth, Linear codes with two or three weights from weakly regular bent functions, IEEE Transactions on Information Theory, 62 (2016), 1166-1176.  doi: 10.1109/TIT.2016.2518678.

[29]

L. C. Washington, Introduction to Cyclotomic Fields, Graduate Texts in Mathematics, 83. Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-1934-7.

[30]

J. Yuan and C. Ding, Secret sharing schemes from three classes of linear codes, IEEE Transactions on Information Theory, 52 (2006), 206-212.  doi: 10.1109/TIT.2005.860412.

[31]

Z. ZhouN. LiC. Fan and T. Helleseth, Linear codes with two or three weights from quadratic bent functions, Designs, Codes and Cryptography, 81 (2016), 283-295.  doi: 10.1007/s10623-015-0144-9.

show all references

References:
[1]

R. AndersonC. DingT. Helleseth and T. Klove, How to build robust shared control systems, Designs, Codes and Cryptography, 15 (1998), 111-124.  doi: 10.1023/A:1026421315292.

[2]

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

[3]

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.

[4] C. Carlet, Boolean Functions for Cryptography and Coding Theory, Cambridge University Press, Cambridge, U.K., 2021.  doi: 10.1017/9781108606806.
[5]

C. CarletC. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Transactions on Information Theory, 51 (2005), 2089-2102.  doi: 10.1109/TIT.2005.847722.

[6]

A. Çeşmelioǧlu and W. Meidl, Bent functions of maximal degree, IEEE Transactions on Information Theory, 58 (2012), 1186-1190.  doi: 10.1109/TIT.2011.2170053.

[7]

A. Çeşmelioǧlu and W. Meidl, A construction of bent functions from plateaued functions, Designs, Codes and Cryptography, 66 (2013), 231-242.  doi: 10.1007/s10623-012-9686-2.

[8]

A. ÇeşmelioǧluW. Meidl and A. Pott, Generalized Maiorana-McFarland class and normality of $p$-ary bent functions, Finite Fields and Their Applications, 24 (2013), 105-117.  doi: 10.1016/j.ffa.2013.06.001.

[9]

A. ÇeşmelioǧluW. Meidl and A. Pott, On the dual of (non)-weakly regular bent functions and self-dual bent functions, Advances in Mathematics of Communications, 7 (2013), 425-440.  doi: 10.3934/amc.2013.7.425.

[10]

C. Ding and H. Niederreiter, Cyclotomic linear codes of order 3, IEEE Transactions on Information Theory, 53 (2007), 2274-2277.  doi: 10.1109/TIT.2007.896886.

[11]

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

[12]

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

[13]

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 3rd ed, Oxford, at the Clarendon Press, 1954.

[14]

T. Helleseth and A. Kholosha, Monomial and quadratic bent functions over the finite fields of odd characteristic, IEEE Transactions on Information Theory, 52 (2006), 2018-2032.  doi: 10.1109/TIT.2006.872854.

[15]

T. Helleseth and A. Kholosha, Crosscorrelation of m-sequences exponential sums bent functions and jacobsthal sums, Cryptography and Communications, 3 (2011), 281-291.  doi: 10.1007/s12095-011-0048-0.

[16]

T. Helleseth and A. Kholosha, New binomial bent functions over the finite fields of odd characteristic, IEEE Transactions on Information Theory, 56 (2010), 4646-4652.  doi: 10.1109/TIT.2010.2055130.

[17]

J. Y. HyunJ. Lee and Y. Lee, Explicit criterions for construction of plateaued functions, IEEE Transactions on Information Theory, 62 (2016), 7555-7565.  doi: 10.1109/TIT.2016.2582217.

[18]

I. M. Isaacs, Character Theory of Finite Groups, AMS Chelsea Publishing, Providence, RI, 2006. doi: 10.1090/chel/359.

[19]

P. V. KumarR. A. Scholtz and L. R. Welch, Generalized bent functions and their properties, J. Combinatorial Theory Ser. A, 40 (1985), 90-107.  doi: 10.1016/0097-3165(85)90049-4.

[20]

N. Li and S. Mesnager, Recent results and problems on constructions of linear codes from cryptographic functions, Cryptogr. Commun., 12 (2020), 965-986.  doi: 10.1007/s12095-020-00435-1.

[21]

S. Mesnager, Linear codes with few weights from weakly regular bent functions based on a generic construction, Cryptography and Communications, 9 (2017), 71-84.  doi: 10.1007/s12095-016-0186-5.

[22]

S. Mesnager, Bent Functions: Fundamentals and Results, Springer, [Cham], 2016. doi: 10.1007/978-3-319-32595-8.

[23]

S. Mesnager, Linear Codes from Functions, Chapter 20 in A Concise Encyclopedia of Coding Theory CRC Press/Taylor and Francis Group (Publisher) London, New York, 2021, 94 pp. doi: 10.1201/9781315147901.

[24]

S. MesnagerF. Özbudak and A. Sınak, Linear codes from weakly regular plateaued functions and their secret sharing schemes, Designs, Codes and Cryptography, 87 (2019), 463-480.  doi: 10.1007/s10623-018-0556-4.

[25]

F. Özbudak and R. M. Pelen, Duals of non-weakly regular bent functions are not weakly regular and generalization to plateaued functions, Finite Fields and Their Applications, 64 (2020), 101668, 16 pp. doi: 10.1016/j.ffa.2020.101668.

[26]

F. Özbudak and R. M. Pelen, Strongly regular graphs arising from non-weakly regular bent functions, Cryptography and Communications, 11 (2019), 1297-1306.  doi: 10.1007/s12095-019-00394-2.

[27]

R. M. Pelen, Studies on Non-Weakly Regular Bent Functions and Related Structures, Thesis (Ph.D.) – Graduate School of Natural and Applied Sciences. Mathematics., Middle East Technical University, 2020.

[28]

C. TangN. LiY. QiZ. Zhou and and T. Helleseth, Linear codes with two or three weights from weakly regular bent functions, IEEE Transactions on Information Theory, 62 (2016), 1166-1176.  doi: 10.1109/TIT.2016.2518678.

[29]

L. C. Washington, Introduction to Cyclotomic Fields, Graduate Texts in Mathematics, 83. Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-1934-7.

[30]

J. Yuan and C. Ding, Secret sharing schemes from three classes of linear codes, IEEE Transactions on Information Theory, 52 (2006), 206-212.  doi: 10.1109/TIT.2005.860412.

[31]

Z. ZhouN. LiC. Fan and T. Helleseth, Linear codes with two or three weights from quadratic bent functions, Designs, Codes and Cryptography, 81 (2016), 283-295.  doi: 10.1007/s10623-015-0144-9.

Table 1.  The weight distribution of $ \mathcal {C}_{C_{j_0}(f)} $ when $ n $ is even
Hamming weight $ a $ Multiplicity $ E_a $
0 1
$ 23^{r-2} $ $ 3^{2r-n-1}+3^{r-\frac{n}{2}-1} $
$ 2(3^{r-2}+3^{\frac{n}{2}-1}) $ $ 23^{2r-n-1}-23^{r-\frac{n}{2}-1} $
$ 2(3^{r-2}-3^{\frac{n}{2}-2}+3^{\frac{n}{2}-1}) $ $ 3^r-3^{2r-n} $
Hamming weight $ a $ Multiplicity $ E_a $
0 1
$ 23^{r-2} $ $ 3^{2r-n-1}+3^{r-\frac{n}{2}-1} $
$ 2(3^{r-2}+3^{\frac{n}{2}-1}) $ $ 23^{2r-n-1}-23^{r-\frac{n}{2}-1} $
$ 2(3^{r-2}-3^{\frac{n}{2}-2}+3^{\frac{n}{2}-1}) $ $ 3^r-3^{2r-n} $
Table 2.  The weight distribution of $ \mathcal {C}_{C_{j_0+2}(f)} $ when $ n $ is odd
Hamming weight $ a $ Multiplicity $ E_a $
0 1
$ 23^{r-2} $ $ 3^{2r-n-1}-1 $
$ 2\left(3^{r-2}+3^{\frac{n-3}{2}}\right) $ $ 3^{r}-23^{2r-n-1}-3^{r-\frac{n+1}{2}} $
$ 2\left(3^{r-2}+23^{\frac{n-3}{2}}\right) $ $ 3^{2r-n-1}+3^{r-\frac{n+1}{2}} $
Hamming weight $ a $ Multiplicity $ E_a $
0 1
$ 23^{r-2} $ $ 3^{2r-n-1}-1 $
$ 2\left(3^{r-2}+3^{\frac{n-3}{2}}\right) $ $ 3^{r}-23^{2r-n-1}-3^{r-\frac{n+1}{2}} $
$ 2\left(3^{r-2}+23^{\frac{n-3}{2}}\right) $ $ 3^{2r-n-1}+3^{r-\frac{n+1}{2}} $
Table 3.  The weight distribution of $ \mathcal {C}_{D_{j_0+2}(f)} $ when $ n $ is even
Hamming weight $ a $ Multiplicity $ E_a $
0 1
$ 23^{r-2} $ $ 23^{2r-n-1}-3^{r-\frac{n}{2}-1}-1 $
$ 2\left(3^{r-2}+3^{\frac{n}{2}-1}\right) $ $ 3^{2r-n-1}+3^{r-\frac{n}{2}-1} $
$ 2\left(3^{r-2}+3^{\frac{n}{2}-2}\right) $ $ 3^r-3^{2r-n} $
Hamming weight $ a $ Multiplicity $ E_a $
0 1
$ 23^{r-2} $ $ 23^{2r-n-1}-3^{r-\frac{n}{2}-1}-1 $
$ 2\left(3^{r-2}+3^{\frac{n}{2}-1}\right) $ $ 3^{2r-n-1}+3^{r-\frac{n}{2}-1} $
$ 2\left(3^{r-2}+3^{\frac{n}{2}-2}\right) $ $ 3^r-3^{2r-n} $
Table 4.  The weight distribution of $ \mathcal {C}_{D_{j_0+1}(f)} $ when $ n $ is odd
Hamming weight $ a $ Multiplicity $ E_a $
0 1
$ 23^{r-2} $ $ 3^{2r-n-1}-1 $
$ 2\left(3^{r-2}+3^{\frac{n-3}{2}}\right) $ $ 3^{r}-23^{2r-n-1}-3^{r-\frac{n+1}{2}} $
$ 2\left(3^{r-2}+23^{\frac{n-3}{2}}\right) $ $ 3^{2r-n-1}+3^{r-\frac{n+1}{2}} $
Hamming weight $ a $ Multiplicity $ E_a $
0 1
$ 23^{r-2} $ $ 3^{2r-n-1}-1 $
$ 2\left(3^{r-2}+3^{\frac{n-3}{2}}\right) $ $ 3^{r}-23^{2r-n-1}-3^{r-\frac{n+1}{2}} $
$ 2\left(3^{r-2}+23^{\frac{n-3}{2}}\right) $ $ 3^{2r-n-1}+3^{r-\frac{n+1}{2}} $
[1]

Ayça Çeşmelioǧlu, Wilfried Meidl, Alexander Pott. On the dual of (non)-weakly regular bent functions and self-dual bent functions. Advances in Mathematics of Communications, 2013, 7 (4) : 425-440. doi: 10.3934/amc.2013.7.425

[2]

Yanfeng Qi, Chunming Tang, Zhengchun Zhou, Cuiling Fan. Several infinite families of p-ary weakly regular bent functions. Advances in Mathematics of Communications, 2018, 12 (2) : 303-315. doi: 10.3934/amc.2018019

[3]

Ayça Çeşmelioğlu, Wilfried Meidl. Bent and vectorial bent functions, partial difference sets, and strongly regular graphs. Advances in Mathematics of Communications, 2018, 12 (4) : 691-705. doi: 10.3934/amc.2018041

[4]

Claude Carlet, Juan Carlos Ku-Cauich, Horacio Tapia-Recillas. Bent functions on a Galois ring and systematic authentication codes. Advances in Mathematics of Communications, 2012, 6 (2) : 249-258. doi: 10.3934/amc.2012.6.249

[5]

Joaquim Borges, Josep Rifà, Victor A. Zinoviev. On $q$-ary linear completely regular codes with $\rho=2$ and antipodal dual. Advances in Mathematics of Communications, 2010, 4 (4) : 567-578. doi: 10.3934/amc.2010.4.567

[6]

Alexander Barg, Arya Mazumdar, Gilles Zémor. Weight distribution and decoding of codes on hypergraphs. Advances in Mathematics of Communications, 2008, 2 (4) : 433-450. doi: 10.3934/amc.2008.2.433

[7]

Alexander A. Davydov, Stefano Marcugini, Fernanda Pambianco. On the weight distribution of the cosets of MDS codes. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021042

[8]

Jacques Wolfmann. Special bent and near-bent functions. Advances in Mathematics of Communications, 2014, 8 (1) : 21-33. doi: 10.3934/amc.2014.8.21

[9]

Chunming Tang, Maozhi Xu, Yanfeng Qi, Mingshuo Zhou. A new class of $ p $-ary regular bent functions. Advances in Mathematics of Communications, 2021, 15 (1) : 55-64. doi: 10.3934/amc.2020042

[10]

Nigel Boston, Jing Hao. The weight distribution of quasi-quadratic residue codes. Advances in Mathematics of Communications, 2018, 12 (2) : 363-385. doi: 10.3934/amc.2018023

[11]

Eimear Byrne. On the weight distribution of codes over finite rings. Advances in Mathematics of Communications, 2011, 5 (2) : 395-406. doi: 10.3934/amc.2011.5.395

[12]

Petr Lisoněk, Layla Trummer. Algorithms for the minimum weight of linear codes. Advances in Mathematics of Communications, 2016, 10 (1) : 195-207. doi: 10.3934/amc.2016.10.195

[13]

Gerardo Vega, Jesús E. Cuén-Ramos. The weight distribution of families of reducible cyclic codes through the weight distribution of some irreducible cyclic codes. Advances in Mathematics of Communications, 2020, 14 (3) : 525-533. doi: 10.3934/amc.2020059

[14]

Bimal Mandal, Aditi Kar Gangopadhyay. A note on generalization of bent boolean functions. Advances in Mathematics of Communications, 2021, 15 (2) : 329-346. doi: 10.3934/amc.2020069

[15]

Claude Carlet, Fengrong Zhang, Yupu Hu. Secondary constructions of bent functions and their enforcement. Advances in Mathematics of Communications, 2012, 6 (3) : 305-314. doi: 10.3934/amc.2012.6.305

[16]

Sihem Mesnager, Fengrong Zhang. On constructions of bent, semi-bent and five valued spectrum functions from old bent functions. Advances in Mathematics of Communications, 2017, 11 (2) : 339-345. doi: 10.3934/amc.2017026

[17]

Ricardo A. Podestá, Denis E. Videla. The weight distribution of irreducible cyclic codes associated with decomposable generalized Paley graphs. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021002

[18]

Lanqiang Li, Shixin Zhu, Li Liu. The weight distribution of a class of p-ary cyclic codes and their applications. Advances in Mathematics of Communications, 2019, 13 (1) : 137-156. doi: 10.3934/amc.2019008

[19]

Chengju Li, Sunghan Bae, Shudi Yang. Some two-weight and three-weight linear codes. Advances in Mathematics of Communications, 2019, 13 (1) : 195-211. doi: 10.3934/amc.2019013

[20]

Dandan Wang, Xiwang Cao, Gaojun Luo. A class of linear codes and their complete weight enumerators. Advances in Mathematics of Communications, 2021, 15 (1) : 73-97. doi: 10.3934/amc.2020044

2021 Impact Factor: 1.015

Metrics

  • PDF downloads (126)
  • HTML views (68)
  • Cited by (0)

Other articles
by authors

[Back to Top]