Theta functions and symmetric weight enumerators for codes over imaginary quadratic fields. (English) Zbl 1397.94124
Summary: In this paper we continue the study of codes over imaginary quadratic fields and their weight enumerators and theta functions. We present new examples of non-equivalent codes over rings of characteristic \(p=2\) and \(p=5\) which have the same theta functions. We also look at a generalization of codes over imaginary quadratic fields, providing examples of non-equivalent pairs with the same theta function for \(p=3\) and \(p=5\).
MSC:
| 94B27 | Geometric methods (including applications of algebraic geometry) applied to coding theory |
| 11H71 | Relations with coding theory |
References:
| [1] | Bachoc C.: Applications of coding theory to the construction of modular lattices. J. Comb. Theory Ser. A 78(1), 92-119 (1997). · Zbl 0876.94053 |
| [2] | Chua K.S.: Codes over \[\rm{GF}(4)\] GF(4) and \[F_2\times 2\]×\[F_22\] and Hermitian lattices over imaginary quadratic fields. Proc. Am. Math. Soc. 133(3), 661-670 (2005). · Zbl 1162.11360 |
| [3] | Dougherty S., Kim J.-L., Lee Y.: Linear codes, hermitian lattices, and finite rings (in press). · Zbl 1359.94688 |
| [4] | Günther A., Nebe G.: Clifford-Weil groups for finite group rings, some examples. Alban. J. Math. 2(3), 185-198 (2008). · Zbl 1210.94110 |
| [5] | Günther A., Nebe G., Rains E.M.: Clifford-Weil groups of quotient representations. Alban. J. Math. 2(3), 159-169 (2008). · Zbl 1210.94111 |
| [6] | Leech J., Sloane N.J.A.: Sphere packing and error-correcting codes. Can. J. Math. 23, 718-745 (1971). · Zbl 0207.52205 |
| [7] | MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. II. North-Holland Mathematical Library, vol. 16, pp. 1-10, 370-762. North-Holland, Amsterdam (1977). · Zbl 0369.94008 |
| [8] | MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. I. North-Holland Mathematical Library, vol. 16, pp. 1-15, 1-369. North-Holland, Amsterdam (1977). · Zbl 0369.94008 |
| [9] | Shaska T., Shor C.: Codes over \[F_{p^2}\] Fp2 and \[F_p \times F_p\] Fp×Fp, lattices, and corresponding theta functions. Adv. Coding Theory Cryptol. 3, 70-80 (2007). · Zbl 1136.11033 |
| [10] | Shaska T., Wijesiri S.: Codes over rings of size four, Hermitian lattices, and corresponding theta functions. Proc. Am. Math. Soc. 136, 849-960 (2008). · Zbl 1133.11042 |
| [11] | Shaska T., Shor C., Wijesiri S.: Codes over rings of size \[p^2\] p2 and lattices over imaginary quadratic fields. Finite Fields Appl. 16(2), 75-87 (2010). · Zbl 1188.11016 |
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