Cyclic codes over the ring $$F_2+uF_2+vF_2$$ F 2
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Cyclic codes over the ring F2 + u F2 + v F2 Karim Samei1 · Mohammad Reza Alimoradi1
Received: 14 September 2016 / Revised: 24 May 2017 / Accepted: 24 May 2017 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017
Abstract In this paper, we study linear and cyclic codes over the ring F2 + u F2 + v F2 . The ring F2 + u F2 + v F2 is the smallest non-Frobenius ring. We characterize the structure of cyclic codes over the ring R = F2 + u F2 + v F2 using of the work Abualrub and Saip (Des Codes Cryptogr 42:273–287, 2007). We study the rank and dual of cyclic codes of odd length over this ring. Specially, we show that the equation |C||C ⊥ | = |R|n does not hold in general for a cyclic code C of length n over this ring. We also obtain some optimal binary codes as the images of cyclic codes over the ring F2 + u F2 + v F2 under a Gray map, which maps Lee weights to Hamming weights. Finally, we give a condition for cyclic codes over R that contains its dual and find quantum codes over F2 from cyclic codes over the ring F2 + u F2 + v F2 . Keywords Frobenius ring · Optimal codes · Quantum codes · Gray map Mathematics Subject Classification 94B15
1 Introduction Cyclic codes are an important class of codes and have been studied over finite fields, chain rings, Galios rings and principal ideals rings. Specially, cyclic codes over finite chain rings have been studied in different manners by numerous authors (Abualrub and Saip 2007; Bonnecaze and Udaya 1999; Greferath and Schmidt 1999; Dinh and López-Permouth 2004; Wolfmann 1999). Recently, cyclic codes over the ring F2 + u F2 + v F2 + uv F2 have been
Communicated by Masaaki Harada.
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Karim Samei [email protected] Mohammad Reza Alimoradi [email protected]
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Department of Mathematics, Bu-Ali Sina university, Hamedan, Iran
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K. Samei, M. R. Alimoradi
considered by Yildiz and Karadeniz (2011), and some good binary codes have been obtained as the images under two Gray maps. Since the ring F2 +u F2 +v F2 +uv F2 is not a finite chain ring, some techniques used in these literatures are different from those in the previous papers. The ring F2 +u F2 +v F2 +uv F2 is a local Frobenius ring. Note that, finite Frobenius rings are significant for coding theoretic purposes because two classical theorems of MacWilliams, the extension theorem and the MacWilliams identities, are valid for finite Frobenius rings (Wood 1999, Theorem 6.3 and Theorem 8.1). Also, a code C of length n over a finite Frobenius ring R and its dual satisfy the following equations (Wood 1999). |C||C ⊥ | = |R|n , (C ⊥ )⊥ = C. But so far, there is not any work on the structure of codes over non-Frobenius rings. In this work, for the first time we study cyclic codes over a non-Frobenius ring. In other words, we study cyclic codes over the non-Frobenius ring R = F2 + u F2 + v F2 , which was introduced by Wood (1999). Specially, we show that the equation |C||C ⊥ | = |R|n does hold if and only if C is a cyclic code of length n over the Frobenius subring F2 + u F2 of F2 + u F2 + v F2 . The organiz
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