There is a one-to-one correspondence between ℓ-quasi-cyclic codes over a finite field F q and linear codes over a ring R = F q [Y ]/(Y m − 1). Using this correspondence, we prove that every ℓ-quasi-cyclic self-dual code of length mℓ over a finite field F q can be obtained by the building-up construction, provided that char (F q ) = 2 or q ≡ 1 (mod 4), m is a prime p, and q is a primitive element of F p . We determine possible weight enumerators of a binary ℓ-quasi-cyclic self-dual code of length pℓ (with p a prime) in terms of divisibility by p. We improve the result of [3] by constructing new binary cubic (i.e., ℓ-quasi-cyclic codes of length 3ℓ) optimal selfdual codes of lengths 30, 36, 42, 48 (Type I), 54 and 66. We also find quasi-cyclic optimal self-dual codes of lengths 40, 50, and 60. When m = 5, we obtain a new 8-quasi-cyclic self-dual [40,20,12] code over F 3 and a new 6-quasi-cyclic self-dual [30,15,10] code over F 4 . When m = 7, we find a new 4-quasi-cyclic self-dual [28,14,9] code over F 4 and a new 6-quasi-cyclic self-dual [42,21,12] code over F 4 .
Abstract. We present two kinds of construction methods for self-dual codes over F 2 + uF 2 . Specially, the second construction (respectively, the first one) preserves the types of codes, that is, the constructed codes from Type II (respectively, Type IV) is also Type II (respectively, Type IV). Every Type II (respectively, Type IV) code over F 2 + uF 2 of free rank larger than three (respectively, one) can be obtained via the second construction (respectively, the first one). Using these constructions, we update the information on self-dual codes over F 2 +uF 2 of length 9 and 10, in terms of the highest minimum (Hamming, Lee, or Euclidean) weight and the number of inequivalent codes with the highest minimum weight.
We prove that the MacWilliams duality holds for bent functions. It enables us to derive the concept of formally self-dual Boolean functions with respect to their near weight enumerators. By using this concept, we prove the Gleason-type theorem on self-dual bent functions. As an application, we provide the total number of (self-dual) bent functions in two and four variables obtaining from formally self-dual Boolean functions.
In a previous paper [13] one of us considered Brauer groups Br(C) and class groups Cl(C) attached to certain monoidal categories C of divisorial R-lattices. That paper showed that the groups arising for a suitable pair of categories C1 ⊆ C2 could be related by a tidy exact sequenceIt was shown that this exact sequence specializes to a number of exact sequences which had formerly been handled separately. At the same time the conventional setting of noetherian normal domains was replaced by that of Krull domains, thus generalizing previous results while also simplifying the proofs. This work was carried out in an affine setting, and one aim of the present paper is to carry these results over to Krull schemes. This will enable us to recover the non-affine version of an exact sequence obtained by Auslander [1, p. 261], as well as to introduce a new, non-affine version of a different sequence derived by the same author [2, Theorem 1].
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