Article

Optimal Self-Dual Codes Over>tex<$ BBF _2times BBF _2$>/tex<With Respect to the Hamming Weight

Dept. of Bus. Inf. Sci., Jobu Univ., Gunma, Japan
IEEE Transactions on Information Theory (Impact Factor: 2.33). 03/2004; 50(2):356 - 358. DOI: 10.1109/TIT.2003.822576
Source: IEEE Xplore

ABSTRACT

In this paper, we study optimal self-dual codes and type IV self-dual codes over the ring F2×F2 of order 4. We give improved upper bounds on minimum Hamming and Lee weights for such codes. Using the bounds, we determine the highest minimum Hamming and Lee weights for such codes of lengths up to 30. We also construct optimal self-dual codes and type IV self-dual codes.

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    • "We can list some related studies on this subject that study codes over chain rings such as the ring of four elements F 2 + uF 2 , the ring of 8 elements F 2 + uF 2 + u 2 F 2 , and a more general chain ring F 2 [u]/u s are presented in [2–4, 6, 11, 13]. Some Euclidean and Hermitian self-dual codes over the ring F 2 [v]/v 2 − v are related to binary self-dual and formally self-dual codes and optimal self-dual binary codes obtained in [5] which inspired the original work of the authors [4]. Gao studied a new generalization of [4] over F p under the restriction v 3 − v in [8]. "
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    ABSTRACT: In this study, we consider linear and especially cyclic codes over the non-chain ring Zp[v]/⟨vp − v⟩ where p is a prime. This is a generalization of the case p = 3. Further, in this work the structure of constacyclic codes are studied as well. This study takes advantage mainly from a Gray map which preserves the distance between codes over this ring and p-ary codes and moreover this map enlightens the structure of these codes. Furthermore, a MacWilliams type identity is presented together with some illustrative examples.
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    • "We can list some related studies on this subject that study codes over chain rings such as the ring of four elements F 2 + uF 2 , the ring of 8 elements F 2 + uF 2 + u 2 F 2 , and a more general chain ring F 2 [u]/u s are presented in [2–4, 6, 11, 13]. Some Euclidean and Hermitian self-dual codes over the ring F 2 [v]/v 2 − v are related to binary self-dual and formally self-dual codes and optimal self-dual binary codes obtained in [5] which inspired the original work of the authors [4]. Gao studied a new generalization of [4] over F p under the restriction v 3 − v in [8]. "

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    • "11, 13, 17, 19, 23 and 29 The code over F p + vF p Gray image; over F p QR 11 (5) 6, 121 3 , 5 [12, 6, 5] 11 formally self-dual QR 11 (7) 8, 121 4 , 7 [16, 8, 7] 11 self-dual QR 11 (19) 20, 121 10 , 13 [40, 20, 13] 11 self-dual QR 13 (3) 8, 169 2 , 4 [8] [4] [4] "
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    ABSTRACT: In this paper quadratic residue codes over the ring Fp+vFpFp+vFp are introduced in terms of their idempotent generators. The structure of these codes is studied and it is observed that these codes enjoy similar properties as quadratic residue codes over finite fields. For the case p=2p=2, Euclidean and Hermitian self-dual families of codes as extended quadratic residue codes are considered and two optimal Hermitian self-dual codes are obtained as examples. Moreover, a substantial number of good p -ary codes are obtained as images of quadratic residue codes over Fp+vFpFp+vFp in the cases where p is an odd prime. These results are presented in tables.
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