Finite Non-chain Ring Research Articles

Quadratic residue codes over the ring 𝔽 p [ u ] / 〈 u m − u 〉 $\mathbb {F}_{p}[u]/\langle u^{m}-u\rangle $ and their Gray images

Let m ≥ 2 be any natural number and let \(\mathcal {R}=\mathbb {F}_{p}+u\mathbb {F}_{p}+u^{2}\mathbb {F}_{p}+\cdots +u^{m-1}\mathbb {F}_{p}\) be a finite non-chain ring, where um = u and p is a prime congruent to 1 modulo (m − 1). In this paper we study quadratic residue codes over the ring \(\mathcal {R}\) and their extensions. A Gray map from \(\mathcal {R}^{n}\) to \((\mathbb {F}_{p}^{m})^{n}\) is defined which preserves self duality of linear codes. As a consequence, we construct self-dual, formally self-dual and self-orthogonal codes over \(\mathbb {F}_{p}\). To illustrate this, several examples of self-dual, self-orthogonal and formally self-dual codes are given. Among others a [9,3,6] linear code over \(\mathbb {F}_{7}\) is constructed which is self-orthogonal as well as nearly MDS. The best known linear code with these parameters (ref. Magma) is not self-orthogonal.

Some results on quadratic residue codes over the ring Fp + vFp + v2Fp + v3Fp

Quadratic residue (QR) codes and their extensions over the finite non-chain ring [Formula: see text] are studied, where [Formula: see text], [Formula: see text] is an odd prime and [Formula: see text]. A class of Gray maps preserving the self-duality of linear codes from [Formula: see text] to [Formula: see text] is given. Under a special Gray map, a self-dual code [Formula: see text] over [Formula: see text], a formally self-dual code [Formula: see text] over [Formula: see text] and a formally self-dual code [Formula: see text] over [Formula: see text] are obtained from extended QR codes.

Secret Sharing Schemes from Linear Codes overFp + vFp

In principle, every linear code can be used to construct a secret sharing scheme. However, determining the access structure of the scheme is a very difficult problem. In this paper, we study MacDonald codes over the finite non-chain ring [Formula: see text], where p is a prime and [Formula: see text]. We provide a method to construct a class of two-weight linear codes over the ring. Then, we determine the access structure of secret sharing schemes based on these codes.

Complete weight enumerators of two classes of linear codes

In this paper, we give the complete weight enumerators of two classes of linear codes over the finite field Fp$\mathbb {F}_{p}$, where p is a prime. These linear codes are the torsion codes of MacDonald codes over the finite non-chain ring Fp+vFp$\mathbb {F}_{p}+v\mathbb {F}_{p}$, where v2 = v. We also employ these linear codes to construct systematic authentication codes with new parameters.

Duadic Codes over the Ring F<sub>q</sub>[u] /＜u<sup>m</sup>-u＞ and Their Gray Images

Let m ≥ 2 be any natural number and let be a finite non-chain ring, where and q is a prime power congruent to 1 modulo (m-1). In this paper we study duadic codes over the ring and their extensions. A Gray map from to is defined which preserves self duality of linear codes. As a consequence self-dual, formally self-dual and self-orthogonal codes over are constructed. Some examples are also given to illustrate this.

Quantum codes from cyclic codes over 𝔽q+v𝔽q+v2𝔽q+v3𝔽q

We give a construction of quantum codes over [Formula: see text] from cyclic codes over a finite non-chain ring [Formula: see text], where [Formula: see text], p is a prime, [Formula: see text] and [Formula: see text].

Linear Codes and (1+<i>uv</i>)-Constacyclic Codes over <i>R</i>[<i>v</i>]/(<i>v</i><sup>2</sup>+<i>v</i>)

In this short correspondence, (1+uv)-constacyclic codes over the finite non-chain ring R[v]/(v2+v) are investigated, where R=F2+uF2 with u2=0. Some structural properties of this class of constacyclic codes are studied. Further, some optimal binary linear codes are obtained from these constacyclic codes.

Note on quasi-twisted codes and an application

Recently, Jia proposed the decompositions and trace representations of quasi-twisted (QT) codes over finite fields (Finite Fields Appl 18:237–257, 2012). The present paper can be viewed as a complementary part of Jia’s work. We investigate some other useful properties of \(\lambda \)-QT codes over finite fields, including the lower Hamming distance bounds, enumerations and searching algorithm for generators. As an interesting application of \(\lambda \)-QT codes over finite fields, we study \(\lambda \)-QT codes over the finite non-chain ring \(\mathbb {F}_q+v\mathbb {F}_q\) briefly.

On skew cyclic codes over F<SUB align="right">3 + vF<SUB align="right">3

In this paper, we study skew cyclic codes over the finite non-chain ring R. We investigate the structural properties of skew polynomial ring of the ring R with respect to a non-trivial automorphism. Further, we prove that skew cyclic codes over R are equivalent to either cyclic codes or quasi-cyclic codes.

Some new binary quasi-cyclic codes from codes over the ring $$\mathbb F _2+u\mathbb F _2+v\mathbb F _2+uv\mathbb F _2$$ F 2 + u F 2 + v F 2 + u v F 2

We consider quasi-cyclic codes over the ring \(\mathbb{F }_2+u\mathbb{F }_2+v\mathbb{F }_2+uv\mathbb{F }_2\), a finite non-chain ring that has been recently studied in coding theory. The Gray images of these codes are shown to be binary quasi-cyclic codes. Using this method we have obtained seventeen new binary quasi-cyclic codes that are new additions to the database of binary quasi-cyclic codes. Moreover, we also obtain a number of binary quasi-cyclic codes with the same parameters as best known binary linear codes that otherwise have more complicated constructions.