In this paper we study n ∏ i=1 Z2i -Additive Cyclic Codes. These codes are identified as Z2n [x]submodules of n ∏ i=1 Z2i [x]/〈x αi − 1〉; αi and i being relatively prime for each i = 1, 2, . . . , n. We first define a n ∏ i=1 Z2i -additive cyclic code of a certain length. We then define the distance between two codewords and the minimum distance of such a code. Moreover we relate these to binar...

The tensor product of one code endowed with the Hamming metric and rank is analyzed. This gives a which naturally inherits sum-rank metric. Specializing to cyclic skew-cyclic code, resulting turns out belong recently introduced family cyclic-skew-cyclic codes. A group theoretical description these codes given, after investigating semilinear isometries in Finally, generalization Roos Hartmann-Tz...

We construct a new Gray map from S to F 3n 2 where S = F2+uF2+uF2, u3 = 1. It is both an isometry and a weight preserving map. It was shown that the Gray image of cyclic code over S is quasi-cyclic codes of index 3 and the Gray image of quasi-cyclic code over S is l-quasi-cyclic code of index 3. Moreover, the skew cyclic and skew quasi-cyclic codes over S introduced and the Gray images of them ...

Let R = Z4[u]/〈u〉 = Z4 + uZ4 + . . . + uZ4 (u = 0) where k ∈ Z satisfies k ≥ 2. For any odd positive integer n, it is known that cyclic codes over R of length n are identified with ideals of the ring R[x]/〈x − 1〉. In this paper, an explicit representation for each cyclic code over R of length n is provided and a formula to count the number of codewords in each code is given. Then a formula to c...

Let C be a Z2Z4-additive code of length n > 3. We prove that if the binary Gray image of C, C = Φ(C), is a 1-perfect nonlinear code, then C cannot be a Z2Z4-cyclic code except for one case of length n = 15. Moreover, we give a parity check matrix for this cyclic code. Adding an even parity check coordinate to a Z2Z4-additive 1-perfect code gives an extended 1-perfect code. We also prove that an...

Usually, it is difficult to determine the weight distribution of an irreducible cyclic code. In this paper, we discuss the case when an irreducible cyclic code has the maximal number of distinct nonzero weights and give a necessary and sufficient condition. In this case, we also obtain a divisible property for the weight of a codeword. Further, we present a necessary and sufficient condition fo...

We show that Han’s bijection when restricted to permutations can be carried out in terms of the cyclic major code and the cyclic inversion code. In other words, it maps a permutation π with a cyclic major code (s1, s2, . . . , sn) to a permutation σ with a cyclic inversion code (s1, s2, . . . , sn). We also show that the fixed points of Han’s map can be characterized by the strong fixed points ...

We consider a q-ary quasi-cyclic code C of length m` and index `, where both m and ` are relatively prime to q. If the constituents of C are cyclic codes, we show that C can also be viewed as a 2-D cyclic code of size m × ` over Fq. If we further assume that m and ` are also coprime to each other, then we easily observe that the code C must be equivalent to a cyclic code. The last fact was prov...

In a recent paper, No, Golomb, Gong, Lee, and Gaal conjectured that certain binary sequences having a simple trace description possess the ideal autocorrelation property. In the present correspondence it is shown that each such sequence is balanced and, moreover, that the dual of the linear cyclic code generated by the sequence and its cyclic shifts, is a triple-error correcting code having the...