doi: 10.3934/amc.2021002
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The weight distribution of irreducible cyclic codes associated with decomposable generalized Paley graphs

FaMAF-CIEM (CONICET), Universidad Nacional de Córdoba, Av. Medina Allende 2144, Ciudad Universitaria, Córdoba (5000), República Argentina

* Corresponding author: Ricardo A. Podestá

Received  February 2020 Revised  February 2021 Early access March 2021

Fund Project: Partially supported by CONICET, FonCyT (BID-PICT 2018-02073) and SECyT-UNC

We use known characterizations of generalized Paley graphs which are Cartesian decomposable to explicitly compute the spectra of the corresponding associated irreducible cyclic codes. As applications, we give reduction formulas for the number of rational points in Artin-Schreier curves defined over extension fields and to the computation of Gaussian periods.

Citation: Ricardo A. Podestá, Denis E. Videla. The weight distribution of irreducible cyclic codes associated with decomposable generalized Paley graphs. Advances in Mathematics of Communications, doi: 10.3934/amc.2021002
References:
[1]

R. AkhtarT. Jackson-HendersonR. KarpmanM. BoggessI. JiménezA. Kinzel and D. Pritikin, On the unitary Cayley graph of a finite ring, Electron. J. Combin., 16 (2009), 117-130.   Google Scholar

[2]

L. D. Baumert and R. J. McEliece, Weights of irreducible cyclic codes, Information and Control, 20 (1972), 158-175.  doi: 10.1016/S0019-9958(72)90354-3.  Google Scholar

[3]

B. Berndt, R. J. Evans and K. Williams, Gauss and Jacobi Sums, Wiley, New York, 1998.  Google Scholar

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D. Cvetkovic, M. Doobs and H. Sachs, Spectra of Graphs, Pure and Applied Mathematics, Academic Press, 1980.  Google Scholar

[5]

C. Ding, The weight distribution of some irreducible cyclic codes, IEEE Trans. Inform. Theory, 55 (2009), 955-960.  doi: 10.1109/TIT.2008.2011511.  Google Scholar

[6]

C. Ding, A class of three-weight and four-weight codes, Lecture Notes in Computer Science, 5557, Springer Verlag, (2009) 34–42. doi: 10.1007/978-3-642-01877-0_4.  Google Scholar

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C. Ding and J. Yang, Hamming weights in irreducible cyclic codes, Discrete Mathematics, 313 (2013), 434-446.  doi: 10.1016/j.disc.2012.11.009.  Google Scholar

[8]

H. Q. DinhC. Li and Q. Yue, Recent progress on weight distributions of cyclic codes over finite fields, J. Algebra Comb. Discrete Struct. Appl., 2 (2015), 39-63.   Google Scholar

[9]

K. Feng and J. Luo, Weight distribution of some reducible cyclic codes, Finite Fields Appl., 14 (2008), 390-409.  doi: 10.1016/j.ffa.2007.03.003.  Google Scholar

[10]

C. D. Godsil and G. F. Royle, Algebraic Graph Theory, Graduate Texts in Mathematics, Springer, 2001. doi: 10.1007/978-1-4613-0163-9.  Google Scholar

[11]

A. Garcia and H. Stichtenoth, Topics in Geometry, Coding Theory and Cryptography, Algebra and Applications, Springer, 2007.  Google Scholar

[12]

R. Hammack, W. Imrich and S. Klavžar, Handbook of Product Graphs, CRC Press 2nd edition, 2011.  Google Scholar

[13]

W. Imrich and S. Klavžar, Product Graphs: Structure and Recognition, Wiley-Interscience, 2000.  Google Scholar

[14]

S. LiS. HuT. Feng and G. Ge, The weight distribution of a class of cyclic codes related to Hermitian forms graphs, IEEE Trans. Inform. Theory, 59 (2013), 3064-3067.  doi: 10.1109/TIT.2013.2242957.  Google Scholar

[15]

C. LiQ. Yue and F. Li, Weight distributions of cyclic codes with respect to pairwise coprime order elements, Finite Fields Appl., 28 (2014), 94-114.  doi: 10.1016/j.ffa.2014.01.009.  Google Scholar

[16]

T. K. Lim and C. Praeger, On Generalised Paley Graphs and their automorphism groups, Michigan Math. J., 58 (2009), 293-308.  doi: 10.1307/mmj/1242071694.  Google Scholar

[17]

R. J. McEliece, Irreducible cyclic codes and Gauss sums, Combinatorics in: Proc. NATO Adv. Study Inst., Breukelen, 1974. Math. Centre Tracts 55, Math. Centrum, Amsterdam, 1974.  Google Scholar

[18]

G. Pearce and C. Praeger, Generalised Paley graphs with a product structure, Ann. Comb., 23 (2019), 171-182.  doi: 10.1007/s00026-019-00423-0.  Google Scholar

[19]

R. A. Podestá and D. E. Videla, The spectra of generalized Paley graphs of $q^{\ell}+1$ powers and applications, preprint, arXiv: 1812.03332. Google Scholar

[20]

R. A. Podestá and D. E. Videla, Spectral properties of generalized Paley graphs and of their associated irreducible cyclic codes, preprint, arXiv: 1908.08097v2. Google Scholar

[21]

R. A. Podestá and D. E. Videla, The Waring's problem over finite fields through generalized Paley graphs, Discrete Mathematics, 344 (2021), 112324. doi: 10.1016/j.disc.2021.112324.  Google Scholar

[22]

A. Rao and N. Pinnawala, A family of two-weight irreducible cyclic codes, IEEE Trans. Inform. Theory, 56 (2010), 2568-2570.  doi: 10.1109/TIT.2010.2046201.  Google Scholar

[23]

G. Sabidussi, Graphs with given group and given graph-theoretical properties, Canadian Journal of Mathematics, 9 (1957), 515-525.  doi: 10.4153/CJM-1957-060-7.  Google Scholar

[24]

G. Sabidussi, Graph multiplication, Mathematische Zeitschrift, 72 (1960), 446-457.  doi: 10.1007/BF01162967.  Google Scholar

[25]

B. Schmidt and C. White, All two weight irreducible cyclic codes, Finite Fields Appl., 8 (2002), 1-17.  doi: 10.1006/ffta.2000.0293.  Google Scholar

[26]

A. Sharma and G. K. Bakshi, The weight distribution of some irreducible cyclic codes, Finite Fields Appl., 18 (2012), 144-159.  doi: 10.1016/j.ffa.2011.07.002.  Google Scholar

[27]

T. Storer., Cyclotomy and Difference Sets, Markham Publishing Co., Chicago, 1967.  Google Scholar

[28]

G. Vega and J. Wolfmann, New classes of 2-weight cyclic codes, Des. Codes Cryptogr., 42 (2007), 327-334.  doi: 10.1007/s10623-007-9038-9.  Google Scholar

[29]

Z. ZhouA. ZhangC. Ding and M. Xiong, The weight enumerator of three families of cyclic codes, IEEE Trans. Inform. Theory, 59 (2013), 6002-6009.  doi: 10.1109/TIT.2013.2262095.  Google Scholar

show all references

References:
[1]

R. AkhtarT. Jackson-HendersonR. KarpmanM. BoggessI. JiménezA. Kinzel and D. Pritikin, On the unitary Cayley graph of a finite ring, Electron. J. Combin., 16 (2009), 117-130.   Google Scholar

[2]

L. D. Baumert and R. J. McEliece, Weights of irreducible cyclic codes, Information and Control, 20 (1972), 158-175.  doi: 10.1016/S0019-9958(72)90354-3.  Google Scholar

[3]

B. Berndt, R. J. Evans and K. Williams, Gauss and Jacobi Sums, Wiley, New York, 1998.  Google Scholar

[4]

D. Cvetkovic, M. Doobs and H. Sachs, Spectra of Graphs, Pure and Applied Mathematics, Academic Press, 1980.  Google Scholar

[5]

C. Ding, The weight distribution of some irreducible cyclic codes, IEEE Trans. Inform. Theory, 55 (2009), 955-960.  doi: 10.1109/TIT.2008.2011511.  Google Scholar

[6]

C. Ding, A class of three-weight and four-weight codes, Lecture Notes in Computer Science, 5557, Springer Verlag, (2009) 34–42. doi: 10.1007/978-3-642-01877-0_4.  Google Scholar

[7]

C. Ding and J. Yang, Hamming weights in irreducible cyclic codes, Discrete Mathematics, 313 (2013), 434-446.  doi: 10.1016/j.disc.2012.11.009.  Google Scholar

[8]

H. Q. DinhC. Li and Q. Yue, Recent progress on weight distributions of cyclic codes over finite fields, J. Algebra Comb. Discrete Struct. Appl., 2 (2015), 39-63.   Google Scholar

[9]

K. Feng and J. Luo, Weight distribution of some reducible cyclic codes, Finite Fields Appl., 14 (2008), 390-409.  doi: 10.1016/j.ffa.2007.03.003.  Google Scholar

[10]

C. D. Godsil and G. F. Royle, Algebraic Graph Theory, Graduate Texts in Mathematics, Springer, 2001. doi: 10.1007/978-1-4613-0163-9.  Google Scholar

[11]

A. Garcia and H. Stichtenoth, Topics in Geometry, Coding Theory and Cryptography, Algebra and Applications, Springer, 2007.  Google Scholar

[12]

R. Hammack, W. Imrich and S. Klavžar, Handbook of Product Graphs, CRC Press 2nd edition, 2011.  Google Scholar

[13]

W. Imrich and S. Klavžar, Product Graphs: Structure and Recognition, Wiley-Interscience, 2000.  Google Scholar

[14]

S. LiS. HuT. Feng and G. Ge, The weight distribution of a class of cyclic codes related to Hermitian forms graphs, IEEE Trans. Inform. Theory, 59 (2013), 3064-3067.  doi: 10.1109/TIT.2013.2242957.  Google Scholar

[15]

C. LiQ. Yue and F. Li, Weight distributions of cyclic codes with respect to pairwise coprime order elements, Finite Fields Appl., 28 (2014), 94-114.  doi: 10.1016/j.ffa.2014.01.009.  Google Scholar

[16]

T. K. Lim and C. Praeger, On Generalised Paley Graphs and their automorphism groups, Michigan Math. J., 58 (2009), 293-308.  doi: 10.1307/mmj/1242071694.  Google Scholar

[17]

R. J. McEliece, Irreducible cyclic codes and Gauss sums, Combinatorics in: Proc. NATO Adv. Study Inst., Breukelen, 1974. Math. Centre Tracts 55, Math. Centrum, Amsterdam, 1974.  Google Scholar

[18]

G. Pearce and C. Praeger, Generalised Paley graphs with a product structure, Ann. Comb., 23 (2019), 171-182.  doi: 10.1007/s00026-019-00423-0.  Google Scholar

[19]

R. A. Podestá and D. E. Videla, The spectra of generalized Paley graphs of $q^{\ell}+1$ powers and applications, preprint, arXiv: 1812.03332. Google Scholar

[20]

R. A. Podestá and D. E. Videla, Spectral properties of generalized Paley graphs and of their associated irreducible cyclic codes, preprint, arXiv: 1908.08097v2. Google Scholar

[21]

R. A. Podestá and D. E. Videla, The Waring's problem over finite fields through generalized Paley graphs, Discrete Mathematics, 344 (2021), 112324. doi: 10.1016/j.disc.2021.112324.  Google Scholar

[22]

A. Rao and N. Pinnawala, A family of two-weight irreducible cyclic codes, IEEE Trans. Inform. Theory, 56 (2010), 2568-2570.  doi: 10.1109/TIT.2010.2046201.  Google Scholar

[23]

G. Sabidussi, Graphs with given group and given graph-theoretical properties, Canadian Journal of Mathematics, 9 (1957), 515-525.  doi: 10.4153/CJM-1957-060-7.  Google Scholar

[24]

G. Sabidussi, Graph multiplication, Mathematische Zeitschrift, 72 (1960), 446-457.  doi: 10.1007/BF01162967.  Google Scholar

[25]

B. Schmidt and C. White, All two weight irreducible cyclic codes, Finite Fields Appl., 8 (2002), 1-17.  doi: 10.1006/ffta.2000.0293.  Google Scholar

[26]

A. Sharma and G. K. Bakshi, The weight distribution of some irreducible cyclic codes, Finite Fields Appl., 18 (2012), 144-159.  doi: 10.1016/j.ffa.2011.07.002.  Google Scholar

[27]

T. Storer., Cyclotomy and Difference Sets, Markham Publishing Co., Chicago, 1967.  Google Scholar

[28]

G. Vega and J. Wolfmann, New classes of 2-weight cyclic codes, Des. Codes Cryptogr., 42 (2007), 327-334.  doi: 10.1007/s10623-007-9038-9.  Google Scholar

[29]

Z. ZhouA. ZhangC. Ding and M. Xiong, The weight enumerator of three families of cyclic codes, IEEE Trans. Inform. Theory, 59 (2013), 6002-6009.  doi: 10.1109/TIT.2013.2262095.  Google Scholar

Table 1.  Weight distribution of $ \mathcal{C} $ with $ p\equiv 2,5,7\pmod 9 $ and $ p>5 $
weight frequency weight frequency
$ w_{0,0} = 0 $ $ A_{0,0}=1 $ $w_{0,2}=p^2-1$ $A_{0,2}=3(\tfrac{p^{2}-1}2)^{2}$
$ w_{1,0}=\tfrac{(p-1)^2}2 $ $ A_{1,0}=3(\tfrac{p^{2}-1}2) $ $w_{0,3}=\tfrac{3(p^{2}-1)}2$ $A_{0,3}=(\tfrac{p^{2}-1}2)^3$
$ w_{2,0}=(p-1)^2 $ $ A_{2,0}=3(\tfrac{p^{2}-1}2)^{2} $ $w_{1,1}=p(p-1)$ $A_{1,1}=6(\tfrac{p^{2}-1}2)^{2}$
$ w_{3,0}=\tfrac{3(p-1)^2}2 $ $ A_{3,0}=(\tfrac{p^{2}-1}2)^3 $ $w_{2,1}=\tfrac{p-1}{2} (3p-1)$ $A_{2,1}=3(\tfrac{p^{2}-1}2)^3$
$ w_{0,1}=\tfrac{p^{2}-1}2 $ $ A_{0,1}=3(\tfrac{p^{2}-1}2) $ $w_{1,2}=\tfrac{p-1}{2} (3p+1)$ $A_{1,2}=3(\tfrac{p^{2}-1}2)^3$
weight frequency weight frequency
$ w_{0,0} = 0 $ $ A_{0,0}=1 $ $w_{0,2}=p^2-1$ $A_{0,2}=3(\tfrac{p^{2}-1}2)^{2}$
$ w_{1,0}=\tfrac{(p-1)^2}2 $ $ A_{1,0}=3(\tfrac{p^{2}-1}2) $ $w_{0,3}=\tfrac{3(p^{2}-1)}2$ $A_{0,3}=(\tfrac{p^{2}-1}2)^3$
$ w_{2,0}=(p-1)^2 $ $ A_{2,0}=3(\tfrac{p^{2}-1}2)^{2} $ $w_{1,1}=p(p-1)$ $A_{1,1}=6(\tfrac{p^{2}-1}2)^{2}$
$ w_{3,0}=\tfrac{3(p-1)^2}2 $ $ A_{3,0}=(\tfrac{p^{2}-1}2)^3 $ $w_{2,1}=\tfrac{p-1}{2} (3p-1)$ $A_{2,1}=3(\tfrac{p^{2}-1}2)^3$
$ w_{0,1}=\tfrac{p^{2}-1}2 $ $ A_{0,1}=3(\tfrac{p^{2}-1}2) $ $w_{1,2}=\tfrac{p-1}{2} (3p+1)$ $A_{1,2}=3(\tfrac{p^{2}-1}2)^3$
Table 2.  Weight distribution of $ \mathcal{C}(516,7^6) $
weight frequency weight frequency
$ w_{0,0,0}=0 $ $ A_{0,0,0}=1 $ $w_{0,2,0}=216$ $A_{0,2,0}=114^2$
$ w_{1,0,0}=96 $ $ A_{1,0,0}=228 $ $w_{0,0,2}=180$ $ A_{0,0,2}=114^2$
$ w_{0,1,0}=108 $ $ A_{0,1,0}=228 $ $ w_{1,1,0}=204$ $A_{1,1,0}=2\cdot 114^2$
$ w_{0,0,1}=90 $ $ A_{0,0,1}=228 $ $w_{1,0,1}=186$ $ A_{1,0,1}=2\cdot 114^2$
$ w_{2,0,0}=192 $ $ A_{2,0,0}=114^2 $ $w_{0,1,1}=198$ $A_{0,1,1}=2\cdot 114^2$
weight frequency weight frequency
$ w_{0,0,0}=0 $ $ A_{0,0,0}=1 $ $w_{0,2,0}=216$ $A_{0,2,0}=114^2$
$ w_{1,0,0}=96 $ $ A_{1,0,0}=228 $ $w_{0,0,2}=180$ $ A_{0,0,2}=114^2$
$ w_{0,1,0}=108 $ $ A_{0,1,0}=228 $ $ w_{1,1,0}=204$ $A_{1,1,0}=2\cdot 114^2$
$ w_{0,0,1}=90 $ $ A_{0,0,1}=228 $ $w_{1,0,1}=186$ $ A_{1,0,1}=2\cdot 114^2$
$ w_{2,0,0}=192 $ $ A_{2,0,0}=114^2 $ $w_{0,1,1}=198$ $A_{0,1,1}=2\cdot 114^2$
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