On the lifted Melas code

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• 作者：Adel Alahmadi ; Hussain Alhazmi ; Tor Helleseth…
• 关键词：Melas code ; Cyclic codes ; Codes over \(\mathbb {Z}_{4}\) ;
• 刊名：Cryptography and Communications
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：8
• 期：1
• 页码：7-18
• 全文大小：176 KB
• 参考文献：1.Grassl, M.: Codetables. www.​codetables.​de
  2.Hammons, R. Jr., Kumar, P.V., Calderbank, R., Sloane, N., Solé, P.: The \(\mathbb {Z}_{4}-\) linearity of Kerdock, Preparata, Goethals and related codes. IEEE Trans. Inform. Theory 40, 301–319 (1994)MathSciNet CrossRef MATH
  3.Helleseth, T., Kumar, P.V.: The algebraic decoding of the \(\mathbb {Z}_{4}-\) linear Goethals code. IEEE Trans. on Inform. Theory 41, 2040–2048 (1995)MathSciNet CrossRef MATH
  4.Nechaev, A.: The Kerdock code in a cyclic form. Discrete Math. Appl. 1, 365 -384 (1991)MathSciNet CrossRef MATH
  5.Pless, V., Qian, Z., Solé, P: Cyclic self dual \(\mathbb {Z}_{4}\) codes. Finite Fields and Their Applications, 48–69 (1997)
  6.Wan, Z.-X.: Quaternary codes. World Scientific, Singapore (1997)MATH
  7.Williams, K.S.: Note on cubics over G F(2 m ) and G F(3 n ),. J. Number Theory 7, 361–365 (1975)MathSciNet CrossRef MATH
  
• 作者单位：Adel Alahmadi (1)
  Hussain Alhazmi (1)
  Tor Helleseth (2)
  Rola Hijazi (1)
  Najat Muthana (1)
  Patrick Solé (3)
  
  1. Math Department, King Abdulaziz University, Jeddah, Saudi Arabia
  2. Department of Informatics, University of Bergen, Bergen, Norway
  3. LTCI, Telecom ParisTech, Paris, France
  
• 刊物类别：Computer Science
• 刊物主题：Coding and Information Theory
  Mathematics of Computing
  
• 出版者：Springer New York
• ISSN：1936-2455

文摘

The binary Melas code is a cyclic code with generator polynomial g(u)=p(u)p(u)∗up> where p(u) is a primitive polynomial of odd degree m≥5 and the ∗ denotes reciprocation. The even-weight subcode of a Melas code has generator polynomial (u+1)g(u) and parameters [2 m −1,2 m −2m−2,6]. This code is lifted to \(\mathbb {Z}_{4}\) and the quaternary code is shown to have parameters [2 m −1,2 m −2m−2,d ub> L ub>≥8], where d ub> L ub> denotes the minimum Lee distance. An algebraic decoding algorithm correcting all errors of Lee weight ≤3 is presented for this code. The Gray map of this quaternary code is a binary code with parameters [2 m+1−2,2 m+1−4m−4,d ub> H ub>≥8] where d ub> H ub> is the minimum Hamming distance. For m=5,7 the minimum distance equals the minimum distance of the best known linear code for the given length and code size.