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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 .

The purpose of this paper is to improve the upper bounds of the minimum distances of self-dual codes over F 5 for lengths [22, 26, 28,[32][33][34][35][36][37][38][39][40]. In particular, we prove that there is no [22,11,9] self-dual code over F 5 , whose existence was left open in 1982. We also show that both the Hamming weight enumerator and the Lee weight enumerator of a putative [24, 12, 10] self-dual code over F 5 are unique. Using the building-up construction, we show that there are exactly nine inequivalent optimal self-dual [18,9,7] codes over F 5 up to the monomial equivalence, and construct one new optimal self-dual [20, 10, 8] code over F 5 and at least 40 new inequivalent optimal self-dual [22,11,8] codes.

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.

Quadratic residue codes have been one of the most important classes of algebraic codes. They have been generalized into duadic codes and quadratic double circulant codes. In this paper we introduce a new subclass of double circulant codes, called duadic double circulant codes, which is a generalization of quadratic double circulant codes for prime lengths. This class generates optimal self-dual codes, optimal linear codes, and linear codes with the best known parameters in a systematic way. We describe a method to construct duadic double circulant codes using 4-cyclotomic cosets and give certain duadic double circulant codes over F 2 , F 3 , F 4 , F 5 , and F 7 . In particular, we find a new ternary self-dual [76, 38,18] code and easily rediscover optimal binary self-dual codes with parameters [66, 33,12], [68, 34,12], [86, 43, 16], and [88, 44, 16] as well as a formally self-dual binary [82, 41,14] code.