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• 作者：Yonglin Cao ; Jian Gao ; Fang-Wei Fu
• 关键词：Semisimple multivariable $$\mathbb {F}_q$$ F q ; linear code ; Semisimple abelian $$\mathbb {F}_q$$ F q ; linear code ; Dual code ; Self ; orthogonal code ; Self ; dual code ; 94B05 ; 94B15 ; 11T71
• 刊名：Designs, Codes and Cryptography
• 出版年：2015
• 出版时间：October 2015
• 年：2015
• 卷：77
• 期：1
• 页码：153-177
• 全文大小：634 KB
• 参考文献：1.Berman S.D.: On the theory of group codes. Cybernnetics 3(1), 25鈥?1 (1969).
  2.Bierbrauer J.: Cyclic additive and quantum stabilzer codes. In: Carlet C., Sunar B. (eds.) Arithmetic of Finite Fields: First International Workshop, Madrid, pp. 276鈥?83 (2007).
  3.Calderbank A., Rains E., Shor P., Sloane N.: Quantum error correction via codes over GF(4). IEEE Trans. Inf. Theory 44, 1369鈥?387 (1998).
  4.Charpin P.: Une g茅n茅ralisation de la construction de Berman des codes de Reed et Muller \(p\) -aires. Commun. Algebra 16(11), 2231鈥?246 (1988).
  5.Cox D., Little J., O鈥橲hea D.: Ideals, Varieties, and Algorithms. Springer, New York (2007).
  6.Danielsen L.E.: Graph-based classification of self-dual codes over finite fields. Adv. Math. Commun. 3, 329鈥?48 (2009).
  7.Danielsen L.E.: On the classification of Hermitian self-dual additive codes over GF(\(9\) ). IEEE Trans. Inf. Theory 58, 5500鈥?511 (2012).
  8.Dey B.K., Rajan B.S.: \(\mathbb{F}_q\) -Linear cyclic codes over \(\mathbb{F}_{q^m}\) : DFT approach. Des. Codes Cryptogr. 34, 89鈥?16 (2005).
  9.Huffman W.C.: Additive cyclic codes over \(\mathbb{F}_4\) . Adv. Math. Commun. 1(4), 427鈥?59 (2007).
  10.Huffman W.C.: Additive cyclic codes over \(\mathbb{F}_4\) . Adv. Math. Commun. 2(3), 309鈥?43 (2008).
  11.Huffman W.C.: Cyclic \(\mathbb{F}_q\) -linear \(\mathbb{F}_{q^t}\) -codes. Int. J. Inf. Coding Theory 1(3), 249鈥?84 (2010).
  12.Huffman W.C.: Self-dual \(\mathbb{F}_q\) -linear \(\mathbb{F}_{q^t}\) -codes with an automorphism of prime order. Adv. Math. Commun. 7(1), 57鈥?0 (2013).
  13.Huffman W.C.: On the theory of \(\mathbb{F}_q\) -linear \(\mathbb{F}_{q^t}\) -codes. Adv. Math. Commun. 7(3), 57鈥?0 (2013).
  14.Mart铆nez-Moro E., Pi帽era-Nicol谩s A., R煤a I.F.: Additive semisimple multivariable codes over \(\mathbb{F}_4\) . Des. Codes Cryptogr. 69, 161鈥?80 (2013).
  15.Mart铆nez-Moro E., R煤a I.F.: Multivariable codes over finite chain rings: serial codes. SIAM J. Discret. Math. 20(4), 947鈥?59 (2006).
  16.Mart铆nez-Moro E., R煤a I.F.: On repeated-root multivariable codes over a finite chain ring. Des. Codes Cryptogr. 45, 219鈥?27 (2007).
  17.Poli A.: Important algebraic calculations for \(n\) -variables polynomial codes. Discret. Math. 56(2鈥?), 255鈥?63 (1985).
  18.Rains E.M.: Nonbinary quantum codes. IEEE Trans. Inf. Theory 45, 1827鈥?832 (1999).
  19.Wan Z.-X.: Lectures on Finite Fields and Galois Rings. World Scientific Publishing Co. Inc., River Edge (2003).
  
• 作者单位：Yonglin Cao (1)
  Jian Gao (2)
  Fang-Wei Fu (2)
  
  1. School of Sciences, Shandong University of Technology, Zibo聽, 255091, Shandong, China
  2. Chern Institute of Mathematics and LPMC, Nankai University, Tianjin聽, 300071, China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Combinatorics
  Coding and Information Theory
  Data Structures, Cryptology and Information Theory
  Data Encryption
  Discrete Mathematics in Computer Science
  Information, Communication and Circuits
  
• 出版者：Springer Netherlands
• ISSN：1573-7586

文摘

Let \(\mathbb {F}_q\) be a finite field of cardinality \(q\), \(l\) a prime number and \(\mathbb {F}_{q^l}\) an extension field of \(\mathbb {F}_q\) with degree \(l\). The structure and canonical form decompositions of semisimple multivariable \(\mathbb {F}_q\)-linear codes over \(\mathbb {F}_{q^l}\) are presented. Enumeration and construction of these codes are then investigated. Especially, dual codes, self-orthogonality and self-duality of semisimple abelian \(\mathbb {F}_q\)-linear codes over \(\mathbb {F}_{q^l}\) are studied. Furthermore, self-dual and \(\gamma \)-self dual semisimple abelian \(\mathbb {F}_q\)-linear codes over \(\mathbb {F}_{q^2}\) are considered. Keywords Semisimple multivariable \(\mathbb {F}_q\)-linear code Semisimple abelian \(\mathbb {F}_q\)-linear code Dual code Self-orthogonal code Self-dual code