Cyclotomic numbers and Jacobi sums, introduced over two centuries ago by Gauss and Jacobi, respectively, are pivotal in number theory and find wide applications in combinatorial designs, coding theory, cryptography, and information theory. The cyclotomic problem, focused on determining all cyclotomic numbers, or equivalently evaluating all Jacobi sums of a given order, has been a subject of extensive research. This paper explores their trivariate counterparts, termed "ternary cyclotomic numbers" and "ternary Jacobi sums", highlighting the fundamental properties that mirror those of the classical cases. We show the ternary versions of Fourier series expansions, two symmetry properties, and a summation equation. We further demonstrate that ternary Jacobi sums, with at least one trivial variable, can be evaluated in terms of classical Jacobi sums of the same order. These properties are established through elementary methods that parallel those utilized in classical cases. Based on these properties, then we offer explicit calculations for all ternary Jacobi sums and ternary cyclotomic numbers of order $ e = 2 $, and near-complete results for order $ e = 3 $, with the exception of the elusive integer $ J_{3}(1, 1, 2) $ for us.
Citation: Zhichao Tang, Xiang Fan. Ternary cyclotomic numbers and ternary Jacobi sums[J]. AIMS Mathematics, 2024, 9(10): 26557-26578. doi: 10.3934/math.20241292
Cyclotomic numbers and Jacobi sums, introduced over two centuries ago by Gauss and Jacobi, respectively, are pivotal in number theory and find wide applications in combinatorial designs, coding theory, cryptography, and information theory. The cyclotomic problem, focused on determining all cyclotomic numbers, or equivalently evaluating all Jacobi sums of a given order, has been a subject of extensive research. This paper explores their trivariate counterparts, termed "ternary cyclotomic numbers" and "ternary Jacobi sums", highlighting the fundamental properties that mirror those of the classical cases. We show the ternary versions of Fourier series expansions, two symmetry properties, and a summation equation. We further demonstrate that ternary Jacobi sums, with at least one trivial variable, can be evaluated in terms of classical Jacobi sums of the same order. These properties are established through elementary methods that parallel those utilized in classical cases. Based on these properties, then we offer explicit calculations for all ternary Jacobi sums and ternary cyclotomic numbers of order $ e = 2 $, and near-complete results for order $ e = 3 $, with the exception of the elusive integer $ J_{3}(1, 1, 2) $ for us.
[1] | C. F. Gauss, Disquisitiones arithmeticae, Springer-Verlag, 1986. https://doi.org/10.1007/978-1-4939-7560-0 |
[2] | C. F. Gauss, Werke, Band II, Georg Olms Verlag, 1973. |
[3] | Y. S. Kim, J. S. Chung, J. S. No, H. Chung, On the autocorrelation distributions of Sidel'nikov sequences, IEEE Trans. Inf. Theory, 51 (2005), 3303–3307. https://doi.org/10.1109/TIT.2005.853310 doi: 10.1109/TIT.2005.853310 |
[4] | C. Ma, L. Zeng, Y. Liu, D. Feng, C. Ding, The weight enumerator of a class of cyclic codes, IEEE Trans. Inf. Theory, 57 (2011), 397–402. https://doi.org/10.1109/TIT.2010.2090272 doi: 10.1109/TIT.2010.2090272 |
[5] | C. Ding, Y. Liu, C. Ma, L. Zeng, The weight distributions of the duals of cyclic codes with two zeros, IEEE Trans. Inf. Theory, 57 (2011), 8000–8006. https://doi.org/10.1109/TIT.2011.2165314 doi: 10.1109/TIT.2011.2165314 |
[6] | Y. Liang, J. Cao, X. Chen, S. Cai, X. Fan, Linear complexity of Ding-Helleseth generalized cyclotomic sequences of order eight, Cryptogr. Commun., 11 (2019), 1037–1056. https://doi.org/10.1007/s12095-018-0343-0 doi: 10.1007/s12095-018-0343-0 |
[7] | L. E. Dickson, Cyclotomy, higher congruences, and waring's problem, Amer. J. Math., 57 (1935), 391–424. https://doi.org/10.2307/2371217 doi: 10.2307/2371217 |
[8] | L. E. Dickson, Cyclotomy and trinomial congruences, Trans. Amer. Math. Soc., 37 (1935), 363–380, 1935. https://doi.org/10.2307/1989714 doi: 10.2307/1989714 |
[9] | L. E. Dickson, Cyclotomy when $e$ is composite, Trans. Amer. Math. Soc., 38 (1935), 187–200. https://doi.org/10.2307/1989680 doi: 10.2307/1989680 |
[10] | M. H. Ahmed, J. Tanti, Cyclotomic numbers and Jacobi sums: a survey, In: Class groups of number fields and related topics, Springer-Verlag, 2020. https://doi.org/10.1007/978-981-15-1514-9_12 |
[11] | J. C. Parnami, M. K. Agrawal, A. R. Rajwade, Jacobi sums and cyclotomic numbers for a finite field, Acta Arith., 41 (1982), 1–13. https://doi.org/10.4064/aa-41-1-1-13 doi: 10.4064/aa-41-1-1-13 |
[12] | S. A. Katre, A. R. Rajwade, Complete solution of the cyclotomic problem in ${\bf{F}}_q$ for any prime modulus $l, \; q = p^\alpha, \; p\equiv 1\; ({ \rm mod}\ l)$, Acta Arith., 45 (1985), 183–199. https://doi.org/10.4064/aa-45-3-183-199 doi: 10.4064/aa-45-3-183-199 |
[13] | L. L. Xia, J. Yang, Cyclotomic problem, Gauss sums and Legendre curve, Sci. China Math., 56 (2013), 1485–1508. https://doi.org/10.1007/s11425-013-4653-6 doi: 10.1007/s11425-013-4653-6 |
[14] | V. V. Acharya, S. A. Katre, Cyclotomic numbers of orders $2l$, $l$ an odd prime, Acta Arith., 69 (1995), 51–74. https://doi.org/10.4064/aa-69-1-51-74 doi: 10.4064/aa-69-1-51-74 |
[15] | D. Shirolkar, S. A. Katre, Jacobi sums and cyclotomic numbers of order $l^2$, Acta Arith., 147 (2011), 33–49. https://doi.org/10.4064/aa147-1-2 doi: 10.4064/aa147-1-2 |
[16] | M. H. Ahmed, J. Tanti, A. Hoque, Complete solution to cyclotomy of order $2l^2$ with prime $l$, Ramanujan J., 53 (2020), 529–550. https://doi.org/10.1007/s11139-019-00182-9 doi: 10.1007/s11139-019-00182-9 |
[17] | M. H. Ahmed, J. Tanti, Complete congruences of Jacobi sums of order $2l^2$ with prime $l$, Ramanujan J., 59 (2022), 967–977. https://doi.org/10.1007/s11139-022-00592-2 doi: 10.1007/s11139-022-00592-2 |
[18] | A. Weil, Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc., 55 (1949), 497–508. https://doi.org/10.1090/S0002-9904-1949-09219-4 doi: 10.1090/S0002-9904-1949-09219-4 |
[19] | L. M. Adleman, C. Pomerance, R. S. Rumely, On distinguishing prime numbers from composite numbers, Ann. Math., 117 (1983), 173–206. https://doi.org/10.2307/2006975 doi: 10.2307/2006975 |
[20] | P. van Wamelen, Jacobi sums over finite fields, Acta Arith., 102 (2002), 1–20. https://doi.org/10.4064/aa102-1-1 doi: 10.4064/aa102-1-1 |
[21] | M. H. Ahmed, J. Tanti, S. Pushp, Computation of Jacobi sums of orders $l^2$ and $2l^2$ with odd prime $l$, Indian J. Pure Appl. Math., 54 (2023), 330–343. https://doi.org/10.1007/s13226-022-00256-3 doi: 10.1007/s13226-022-00256-3 |
[22] | K. H. Leung, S. L. Ma, B. Schmidt, New Hadamard matrices of order $4p^2$ obtained from Jacobi sums of order 16, J. Combin. Theory, 113 (2006), 822–838. https://doi.org/10.1016/j.jcta.2005.07.011 doi: 10.1016/j.jcta.2005.07.011 |
[23] | A. Rojas-León, On a generalization of Jacobi sums, Finite Fields Appl., 77 (2022), 101944. https://doi.org/10.1016/j.ffa.2021.101944 doi: 10.1016/j.ffa.2021.101944 |
[24] | S. A. Katre, A. R. Rajwade, Resolution of the sign ambiguity in the determination of the cyclotomic numbers of order $4$ and the corresponding Jacobsthal sum, Math. Scand., 60 (1987), 52–62. https://doi.org/10.7146/math.scand.a-12171 doi: 10.7146/math.scand.a-12171 |
[25] | T. Storer, On the unique determination of the cyclotomic numbers for Galois fields and Galois domains, J. Comb. Theory, 2 (1967), 296–300. http://doi.org/10.1016/S0021-9800(67)80031-0 doi: 10.1016/S0021-9800(67)80031-0 |
[26] | B. C. Berndt, R. J. Evans, K. S. Williams, Gauss and Jacobi sums, John Wiley & Sons, Inc., 1998. |