Optimal binary codes derived from $$\mathbb {F}_{2} \mathbb {F}_4$$ F 2 F 4 -additive cyclic codes
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Optimal binary codes derived from F2 F4 -additive cyclic codes Taher Abualrub1 · Nuh Aydin2 · Ismail Aydogdu3 Received: 23 January 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract In this paper, we study the algebraic structure of additive cyclic codes over the alphabet Fr2 × Fs4 = Fr2 Fs4 , where r and s are non-negative integers, F2 = GF(2) and F4 = GF(4) are the finite fields of 2 and 4 elements, respectively. We determine generator polynomials for F2 F4 -additive cyclic codes. We also introduce a linear map W that depends on the trace map T to relate these codes to binary linear codes over F2 . Further, we investigate the duals of F2 F4 -additive cyclic codes. We show that the dual of any F2 F4 -additive cyclic code is another F2 F4 -additive cyclic code. Using the mapping W , we provide examples of F2 F4 -additive cyclic codes whose binary images have optimal parameters. We also consider additive cyclic codes over F4 and give some examples of optimal parameter quantum codes over F4 . Keywords F2 F4 -additive cyclic codes · Duality · Quantum codes · Optimal codes Mathematics Subject Classification 94B05 · 94B60
1 Introduction Cyclic codes are a very important and a special family of linear codes for both theoretical and practical reasons. They have a rich algebraic structure and many of the best known codes are cyclic codes. Cyclic codes over finite fields were first introduced
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Ismail Aydogdu [email protected] Taher Abualrub [email protected] Nuh Aydin [email protected]
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Department of Mathematics and Statistics, American University of Sharjah, Sharjah, UAE
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Department of Mathematics and Statistics, Kenyon College, Gambier, OH, USA
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Department of Mathematics, Yildiz Technical University, Istanbul, Turkey
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T. Abualrub et al.
by E. Prange in 1957 and 1959 with two Air Force Cambridge Research Laboratory reports. Later, extensive results about cyclic codes over finite fields were given [7,14– 16]. The study of codes over finite rings increased dramatically after the discovery of connections between certain binary codes and some linear codes over Z4 presented in [12]. This paper was a milestone study for the researchers and motivated them to study linear and cyclic codes over different rings [6,10,13,18]. Further, this research expanded to include codes over mixed alphabets as well. For example, Z2 Z4 -additive codes were introduced in [3] as a generalization of linear codes over Z2 and linear codes over Z4 . Abualrub et al. studied Z2 Z4 -additive cyclic codes in [1], while Shi et al. generalized the study of Z2 Z4 -additive codes to the class of Z p Z pk -additive codes in [17]. Additive cyclic codes over the finite field GF(4) were first introduced by Calderbank et al. in [5]. Additive codes over GF(4) are an interesting and important type of codes because of their applications in constructing quantum codes. An additive code over GF(4) of length n is a subgroup of GF(4)n under addition. Hence, any additive code over GF(4) is a k-dimensional GF(2)-
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