Lingyu Diao; Jian Gao

Additive cyclic codes of length (α,β) over ℤ<SUB align="right">pℤ<SUB align="right">p[u] can be viewed as ℤ<SUB align="right">p[u][x]-submodules of ℤ<SUB align="right">p[x]/(x<SUP align="right">α</SUP> − 1) × ℤ<SUB align="right">p[u][x]/(x<SUP align="right">β</SUP> − 1), where ℤ<SUB align="right">p[u] = ℤ<SUB align="right">p + uℤ<SUB align="right">p, u<SUP align="right">2</SUP> = 0. In this paper, we determine the generator polynomials and the minimal generating sets of this family of codes as ℤ<SUB align="right">p[u]-submodules of ℤ<SUB align="right">p[x] / (x<SUP align="right">α</SUP>−1) × ℤ<SUB align="right">p[u][x] / (x<SUP align="right">β</SUP>−1). Further, we also determine the generator polynomials of the dual codes of ℤ<SUB align="right">pℤ<SUB align="right">p[u]-additive cyclic codes. Moreover, some binary quantum codes are constructed by additive cyclic codes over ℤ<SUB align="right">2ℤ<SUB align="right">2[u].]]>

E. Bellini; M. Sala

Given a binary nonlinear code, we provide a deterministic algorithm to compute its weight and distance distribution, and in particular, its minimum weight and its minimum distance, which takes advantage of fast Fourier techniques. This algorithm's performance is similar to that of best-known algorithms for the average case, while it is especially efficient for codes with low information rate. We provide complexity estimates for several cases of interest.]]>

Mokshi Goyal; Madhu Raka

Let f(u) be a polynomial of degree m, m ≥ 2, which splits into distinct linear factors over a finite field 𝔽<SUB align="right">q. Let ℛ = 𝔽<SUB align="right">q[u] / ⟨ f(u) ⟩ be a finite non-chain ring. In this paper, we study duadic codes, their extensions and triadic codes over the ring ℛ. A Gray map from ℛ<SUP align="right">n</SUP> to (𝔽<SUB align="right">q <SUP align="right">m</SUP>)<SUP align="right">n</SUP> is defined which preserves self-duality of linear codes. As a consequence, self-dual, isodual, self-orthogonal and complementary dual(LCD) codes over 𝔽 <SUB align="right">q are constructed. Some examples are also given to illustrate this.]]>

Ted Hurley; Donny Hurley

The unit-derived method in coding theory is shown to be a unique optimal scheme for constructing and analysing codes. In many cases, efficient and practical decoding methods are produced. Codes with efficient decoding algorithms at maximal distances possible are derived from unit schemes. In particular unit-derived codes from Vandermonde or Fourier matrices are particularly commendable giving rise to mds codes of varying rates with practical and efficient decoding algorithms. For a given rate and given error correction capability, explicit codes with efficient error correcting algorithms are designed to these specifications. An explicit constructive proof with an efficient decoding algorithm is given for Shannon's theorem. For a given finite field, codes are constructed which are 'optimal' for this field.]]>

R. Dastbasteh; H. Mousavi; T. Abualrub; N. Aydin; J. Haghighat

In this paper, we study skew cyclic codes with arbitrary length over the ring R = 𝔽<SUB align="right">p + u𝔽<SUB align="right">p where p is an odd prime and u<SUP align="right">2</SUP> = 0. We characterise all skew cyclic codes of length n as left R[x;θ]-submodules of R<SUB align="right">n = R[x;θ] / ⟨x<SUP align="right">n</SUP> − 1⟩. We find all generator polynomials for these codes and describe their minimal spanning sets. Moreover, an encoding algorithm is presented for skew cyclic codes over the ring R. Finally, based on the theory we developed in this paper, we provide examples of codes with good parameters over F<SUB align="right">p with different odd primes p: In fact, example 6 in our paper is a new ternary code in the class of quasi-twisted codes. We also present several examples of optimal codes.]]>