Additive semisimple multivariable codes over \(\mathbb{F}_4\). (English) Zbl 1291.94207
Additive codes over the field over order 4 are codes which are a subgroup of the ambient space. These codes have applications to quantum computing. The authors study additive multivariable codes in this setting which are a generalization of cyclic codes. Specifically, they study codes which are ideals in the ring \(F_4 [x_1,x_2,\dots,x_r] / \langle t_1(x_1),t_2(x_2), \dots, t_r(x_r) \rangle\). Using the semisimple structure of the ring, they give the structure of these codes in the case where the polynomials have no repeated roots. The structure of abelian codes, namely those where \(t_i(x_i) = x_i^{n_i}-1\) are also described. Additionally, they characterize the non-trivial abelian semisimple codes that are self-dual.
Reviewer: Steven T. Dougherty (Scranton)
MSC:
| 94B60 | Other types of codes |
| 11T71 | Algebraic coding theory; cryptography (number-theoretic aspects) |
| 13M10 | Polynomials and finite commutative rings |
| 81P70 | Quantum coding (general) |
References:
| [1] | Berman S.D.: On the theory of group codes. Cybernetics 3(1), 25-31 (1969) · Zbl 0216.55803 · doi:10.1007/BF01072842 |
| [2] | Calderbank A., Rains E., Shor P., Sloane N.: Quantum error correction via codes over GF(4). IEEE Trans. Inf. Theory 44, 1369-1387 (1998) · Zbl 0982.94029 · doi:10.1109/18.681315 |
| [3] | Charpin P.: Une généralisation de la construction de Berman des codes de Reed et Muller p-aires. Commun. Algebra 16(11), 2231-2246 (1988) · Zbl 0649.94013 · doi:10.1080/00927878808823689 |
| [4] | Cox D., Little J., O’Shea D.: Ideals, Varieties, and Algorithms. Springer, New York (2007) · Zbl 1118.13001 · doi:10.1007/978-0-387-35651-8 |
| [5] | Dey B., Rajan \[B.: {\mathbb{F}_q} \] -linear cyclic codes over \[{\mathbb{F}_q^m} \] . Des. Codes Cryptogr. 34, 89-116 (2005) · Zbl 1070.94028 · doi:10.1007/s10623-003-4196-x |
| [6] | Grassl M.: Bounds on the minimum distance of linear codes and quantum codes. http://www.codetables.d. Accessed 2012. · Zbl 1145.94300 |
| [7] | Huffman W.C.: Additive cyclic codes over \[{\mathbb{F}_4} \] . Adv. Math. Commun. 1(4), 427-459 (2007) · Zbl 1194.94199 · doi:10.3934/amc.2007.1.427 |
| [8] | Huffman W.C.: Additive cyclic codes over \[{\mathbb{F}_4} \] . Adv. Math. Commun. 2(3), 309-343 (2008) · Zbl 1207.94080 · doi:10.3934/amc.2008.2.309 |
| [9] | Martínez-Moro E., Rúa I.F.: Multivariable codes over finite chain rings: serial codes. SIAM J. Discret. Math. 20(4), 947-959 (2006) · Zbl 1140.11062 · doi:10.1137/050632208 |
| [10] | Martínez-Moro E., Rúa I.F.: On repeated-root multivariable codes over a finite chain ring. Des. Codes Cryptogr. 45, 219-227 (2007) · Zbl 1196.94088 · doi:10.1007/s10623-007-9114-1 |
| [11] | Peterson W.W., Weldon J.E.J.: Error-Correcting Codes, 2nd edn. MIT Press, Cambridge, MA (1972). · Zbl 0251.94007 |
| [12] | Poli A.: Important algebraic calculations for n-variables polynomial codes. Discret. Math. 56(2-3), 255-263 (1985) · Zbl 0588.94007 · doi:10.1016/0012-365X(85)90032-9 |
| [13] | Poli A., Huguet L.: Error Correcting Codes. Prentice Hall International, Hemel Hempstead (1992). · Zbl 0875.68263 |
| [14] | Shor P.: Scheme for reducing decoherence in quantum memory. Phys. Rev. A 52 (1995). |
| [15] | Steane A.: Simple quantum error correcting codes. Phys. Rev. Lett. 77, 793-797 (1996) · Zbl 0944.81505 · doi:10.1103/PhysRevLett.77.793 |
| [16] | Stein W.A., et al.: Sage Mathematics Software (Version 4.7.2). The Sage Development Team (2011). http://www.sagemath.org. Accessed 2012. |
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