Symplectic spread-based generalized Kerdock codes

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• 作者：S. González ; C. Martínez and I. F. Rúa
• 关键词：(Generalized) Kerdock code ; Linear presentation of codes ; Galois ring ; Symplectic spread ; Finite (Pre)Semifield
• 刊名：Designs, Codes and Cryptography
• 出版年：2007
• 出版时间：February, 2007
• 年：2007
• 卷：42
• 期：2
• 页码：213-226
• 全文大小：275 KB
• 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

Kerdock codes (Kerdock, Inform Control 20:182–187, 1972) are a well-known family of non-linear binary codes with good parameters admitting a linear presentation in terms of codes over the ring \mathbbZ₄{\mathbb{Z}_{4}} (see Nechaev, Diskret Mat 1:123–139, 1989; Hammons et al., IEEE Trans Inform Theory 40:301–319, 1994). These codes have been generalized in different directions: in Calderbank et al. (Proc Lond Math Soc 75:436–480, 1997) a symplectic construction of non-linear binary codes with the same parameters of the Kerdock codes has been given. Such codes are not necessarily equivalent. On the other hand, in Kuzmin and Nechaev (Russ Math Surv 49(5), 1994) the authors give a family of non-linear codes over the finite field F of q = 2 l elements, all of them admitting a linear presentation over the Galois Ring R of cardinality q 2 and characteristic 22. The aim of this article is to merge both approaches, obtaining in this way new families of non-linear codes over F that can be presented as linear codes over the Galois Ring R. The construction uses symplectic spreads.