We prove that the cyclic and constacyclic codes constructed by Grassl Rötteler in International Symposium on Information Theory (ISIT), pp. 1104–1108 (2015) are generalised Reed–Solomon codes. This note can be considered as an addendum to pp (2015). It also appendix Ball Vilar IEEE Trans Inform 68:3796–3805, (2022) where Conjecture 11 of (2015), which was stated for Grassl–Rötteler codes, is pr...

The famous Barnes–Wall lattices can be obtained by applying Construction D to a chain of Reed–Muller codes. By ${\text {D}}^{\text {(cyc)}}$ extended cyclic codes sandwiched between codes, Hu and Nebe (J. London Math. Soc. <xref ref-t...

We consider linear cyclic codes with the locality property, or locally recoverable codes (LRC codes). A family of LRC codes that generalize the classical construction of Reed-Solomon codes was constructed in a recent paper by I. Tamo and A. Barg (IEEE Trans. Inform. Theory, no. 8, 2014). In this paper we focus on optimal cyclic codes that arise from this construction. We give a characterization...

We study ΘS−cyclic codes over the family of rings Ak. We characterize ΘS−cyclic codes in terms of their binary images. A family of Hermitian inner-products is defined and we prove that if a code is ΘS−cyclic then its Hermitian dual is also ΘS−cyclic. Finally, we give constructions of ΘS−cyclic codes. ∗Corresponding author. email: [email protected]

Abstract. The problem of identifying whether the family of cyclic codes is asymptotically good or not is a long-standing open problem in the field of coding theory. It is known in the literature that some families of cyclic codes such as BCH codes and Reed-Solomon codes are asymptotically bad, however in general the answer to this question is not known. A recent result by Nelson and Van Zwam sh...

In analogy to cyclic codes, we study linear codes over finite fields obtained from left ideals in a quotient ring of a (non commutative) skew polynomial ring. The paper shows how existence and properties of such codes are linked to arithmetic properties of skew polynomials. This class of codes is a generalization of the θ-cyclic codes discussed in [1]. However θ-cyclic codes are performant repr...

In this paper, we study the construction of cyclic DNA codes by cyclic codes over the finite chain ring     2 4 1 F u u  . First, we establish a 1-1 correspondence  between DNA pairs and the 16 elements of the ring     2 4 1 F u u  . Considering the biology features of DNA codes, we investigate the structure and properties of selfreciprocal complement cyclic codes. Hamming minimum di...

As an example, take F2, n = 5 and g(x) = x2 + x + 1. The code consists of the 8 codewords 0 · g(x), . . . , (x2 + x + 1) · g(x). Equivalently, we can identify every polynomial with its vector of coefficients to get a codeword in F2 . Verify that a polynomial code is linear and has dimension k = n −m. Also, check that if g(x) = ∑n−k i=0 gix i is the generator polynomial, then an n× k generating ...

In this paper, we construct quantum synchronizable codes (QSCs) based on the sum and intersection of cyclic codes. Further, infinite families of QSCs are obtained from BCH and duadic codes. Moreover, we show that the work of Fujiwara [7] can be generalized to repeated root cyclic codes (RRCCs) such that QSCs are always obtained, which is not the case with simple root cyclic codes. The usefulnes...