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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. Akhtar, T. Jackson-Henderson, R. Karpman, M. Boggess, I. Jiménez, A. Kinzel and D. Pritikin, On the unitary Cayley graph of a finite ring, Electron. J. Combin., 16 (2009), 117-130. [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. [3] B. Berndt, R. J. Evans and K. Williams, Gauss and Jacobi Sums, Wiley, New York, 1998. [4] D. Cvetkovic, M. Doobs and H. Sachs, Spectra of Graphs, Pure and Applied Mathematics, Academic Press, 1980. [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. [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. [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. [8] H. Q. Dinh, C. 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. [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. [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. [11] A. Garcia and H. Stichtenoth, Topics in Geometry, Coding Theory and Cryptography, Algebra and Applications, Springer, 2007. [12] R. Hammack, W. Imrich and S. Klavžar, Handbook of Product Graphs, CRC Press 2nd edition, 2011. [13] W. Imrich and S. Klavžar, Product Graphs: Structure and Recognition, Wiley-Interscience, 2000. [14] S. Li, S. Hu, T. 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. [15] C. Li, Q. 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. [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. [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. [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. [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. [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. [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. [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. [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. [24] G. Sabidussi, Graph multiplication, Mathematische Zeitschrift, 72 (1960), 446-457.  doi: 10.1007/BF01162967. [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. [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. [27] T. Storer., Cyclotomy and Difference Sets, Markham Publishing Co., Chicago, 1967. [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. [29] Z. Zhou, A. Zhang, C. 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.

show all references

##### References:
 [1] R. Akhtar, T. Jackson-Henderson, R. Karpman, M. Boggess, I. Jiménez, A. Kinzel and D. Pritikin, On the unitary Cayley graph of a finite ring, Electron. J. Combin., 16 (2009), 117-130. [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. [3] B. Berndt, R. J. Evans and K. Williams, Gauss and Jacobi Sums, Wiley, New York, 1998. [4] D. Cvetkovic, M. Doobs and H. Sachs, Spectra of Graphs, Pure and Applied Mathematics, Academic Press, 1980. [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. [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. [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. [8] H. Q. Dinh, C. 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. [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. [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. [11] A. Garcia and H. Stichtenoth, Topics in Geometry, Coding Theory and Cryptography, Algebra and Applications, Springer, 2007. [12] R. Hammack, W. Imrich and S. Klavžar, Handbook of Product Graphs, CRC Press 2nd edition, 2011. [13] W. Imrich and S. Klavžar, Product Graphs: Structure and Recognition, Wiley-Interscience, 2000. [14] S. Li, S. Hu, T. 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. [15] C. Li, Q. 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. [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. [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. [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. [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. [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. [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. [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. [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. [24] G. Sabidussi, Graph multiplication, Mathematische Zeitschrift, 72 (1960), 446-457.  doi: 10.1007/BF01162967. [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. [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. [27] T. Storer., Cyclotomy and Difference Sets, Markham Publishing Co., Chicago, 1967. [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. [29] Z. Zhou, A. Zhang, C. 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.
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 $w_{0,0} = 0$ $A_{0,0}=1$ $w_{0,2}=p^2-1$ $A_{0,2}=3(\tfrac{p^{2}-1}2)^{2}$ $w_{1,0}=\tfrac{(p-1)^2}2$ $A_{1,0}=3(\tfrac{p^{2}-1}2)$ $w_{0,3}=\tfrac{3(p^{2}-1)}2$ $A_{0,3}=(\tfrac{p^{2}-1}2)^3$ $w_{2,0}=(p-1)^2$ $A_{2,0}=3(\tfrac{p^{2}-1}2)^{2}$ $w_{1,1}=p(p-1)$ $A_{1,1}=6(\tfrac{p^{2}-1}2)^{2}$ $w_{3,0}=\tfrac{3(p-1)^2}2$ $A_{3,0}=(\tfrac{p^{2}-1}2)^3$ $w_{2,1}=\tfrac{p-1}{2} (3p-1)$ $A_{2,1}=3(\tfrac{p^{2}-1}2)^3$ $w_{0,1}=\tfrac{p^{2}-1}2$ $A_{0,1}=3(\tfrac{p^{2}-1}2)$ $w_{1,2}=\tfrac{p-1}{2} (3p+1)$ $A_{1,2}=3(\tfrac{p^{2}-1}2)^3$
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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