Additive semisimple multivariable codes over ${\mathbb{F}_4}$

详细信息    查看全文

• 作者：E. Martínez-Moro (1)
  A. Pi?era-Nicolás (1)
  I. F. Rúa (2)
  
• 关键词：Additive multivariable codes ; Abelian codes ; Quantum codes ; Duality ; 11T61 ; 94B99 ; 81P70 ; 13M10
• 刊名：Designs, Codes and Cryptography
• 出版年：2013
• 出版时间：November 2013
• 年：2013
• 卷：69
• 期：2
• 页码：161-180
• 全文大小：489KB
• 参考文献：1. Berman S.D.: On the theory of group codes. Cybernetics 3(1), 25-1 (1969) CrossRef
  2. Calderbank A., Rains E., Shor P., Sloane N.: Quantum error correction via codes over GF(4). IEEE Trans. Inf. Theory 44, 1369-387 (1998) CrossRef
  3. Charpin P.: Une généralisation de la construction de Berman des codes de Reed et Muller / p-aires. Commun. Algebra 16(11), 2231-246 (1988) CrossRef
  4. Cox D., Little J., O’Shea D.: Ideals, Varieties, and Algorithms. Springer, New York (2007) CrossRef
  5. Dey B., Rajan B.: ${\mathbb{F}_q}$ -linear cyclic codes over ${\mathbb{F}_q^m}$ . Des. Codes Cryptogr. 34, 89-16 (2005) CrossRef
  6. Grassl M.: Bounds on the minimum distance of linear codes and quantum codes. http://www.codetables.d. Accessed 2012.
  7. Huffman W.C.: Additive cyclic codes over ${\mathbb{F}_4}$ . Adv. Math. Commun. 1(4), 427-59 (2007) CrossRef
  8. Huffman W.C.: Additive cyclic codes over ${\mathbb{F}_4}$ . Adv. Math. Commun. 2(3), 309-43 (2008) CrossRef
  9. Martínez-Moro E., Rúa I.F.: Multivariable codes over finite chain rings: serial codes. SIAM J. Discret. Math. 20(4), 947-59 (2006) CrossRef
  10. Martínez-Moro E., Rúa I.F.: On repeated-root multivariable codes over a finite chain ring. Des. Codes Cryptogr. 45, 219-27 (2007) CrossRef
  11. Peterson W.W., Weldon J.E.J.: Error-Correcting Codes, 2nd edn. MIT Press, Cambridge, MA (1972).
  12. Poli A.: Important algebraic calculations for / n-variables polynomial codes. Discret. Math. 56(2-), 255-63 (1985) CrossRef
  13. Poli A., Huguet L.: Error Correcting Codes. Prentice Hall International, Hemel Hempstead (1992).
  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, 793-97 (1996) CrossRef
  16. Stein W.A., et?al.: Sage Mathematics Software (Version 4.7.2). The Sage Development Team (2011). http://www.sagemath.org. Accessed 2012.
  
• 作者单位：E. Martínez-Moro (1)
  A. Pi?era-Nicolás (1)
  I. F. Rúa (2)
  
  1. Institute of Mathematics (IMUVa) and Applied Mathematics Department, Universidad de Valladolid, Valladolid, Spain
  2. Departamento de Matemáticas, Universidad de Oviedo, Oviedo, Spain
  

文摘

The structure of additive multivariable codes over ${\mathbb{F}_4}$ (the Galois field with 4 elements) is presented. The semisimple case (i.e., when the defining polynomials of the code have no repeated roots) is specifically addressed. These codes extend in a natural way the abelian codes, of which additive cyclic codes of odd length are a particular case. Duality of these codes is also studied.