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Due to their wide applications in consumer electronics, data storage systems and communication systems, cyclic codes have been an interesting subject of study in recent years. The construction of optimal cyclic codes over finite fields is important as they have maximal minimum distance once the length and dimension are given. In this paper, we present two classes of new optimal ternary cyclic codes \begin{document}$ \mathcal{C}_{(2,v)} $\end{document} by using monomials \begin{document}$ x^2 $\end{document} and \begin{document}$ x^v $\end{document} for some suitable \begin{document}$ v $\end{document} and explain the novelty of the codes. Furthermore, the weight distribution of \begin{document}$ \mathcal{C}_{(2,v)}^{\perp} $\end{document} for \begin{document}$ v = \frac{3^{m}-1}{2}+2(3^{k}+1) $\end{document} is determined.

For an odd prime \begin{document}$ p $\end{document} and positive integers \begin{document}$ m $\end{document} and \begin{document}$ \ell $\end{document} , let \begin{document}$ \mathbb{F}_{p^m} $\end{document} be the finite field with \begin{document}$ p^{m} $\end{document} elements and \begin{document}$ R_{\ell,m} = \mathbb{F}_{p^m}[v_1,v_2,\dots,v_{\ell}]/\langle v^{2}_{i}-1, v_{i}v_{j}-v_{j}v_{i}\rangle_{1\leq i, j\leq \ell} $\end{document} . Thus \begin{document}$ R_{\ell,m} $\end{document} is a finite commutative non-chain ring of order \begin{document}$ p^{2^{\ell} m} $\end{document} with characteristic \begin{document}$ p $\end{document} . In this paper, we aim to construct quantum codes from skew constacyclic codes over \begin{document}$ R_{\ell,m} $\end{document} . First, we discuss the structures of skew constacyclic codes and determine their Euclidean dual codes. Then a relation between these codes and their Euclidean duals has been obtained. Finally, with the help of a duality-preserving Gray map and the CSS construction, many MDS and better non-binary quantum codes are obtained as compared to the best-known quantum codes available in the literature.

Based on solutions of certain equations over finite yields, a necessary and sufficient condition for the quinary negacyclic codes with parameters \begin{document}$ [\frac{5^m-1}{2},\frac{5^m-1}{2}-2m,4] $\end{document} to have generator polynomial \begin{document}$ m_{\alpha^3}(x)m_{\alpha^e}(x) $\end{document} is provided. Several classes of new optimal quinary negacyclic codes with the same parameters are constructed by analyzing irreducible factors of certain polynomials over finite fields. Moreover, several classes of new optimal quinary negacyclic codes with these parameters and generator polynomial \begin{document}$ m_{\alpha}(x)m_{\alpha^e}(x) $\end{document} are also presented.

A new method is used to resolve a long-standing conjecture of Niho concerning the crosscorrelation spectrum of a pair of maximum length linear recursive sequences of length $2^{2m}-1$ with relative decimation $d=2^{m+2}-3$ , where $m$ is even. The result indicates that there are at most five distinct crosscorrelation values. Equivalently, the result indicates that there are at most five distinct values in the Walsh spectrum of the power permutation $f(x)=x^{d}$ over a finite field of order $2^{2m}$ and at most five distinct nonzero weights in the cyclic code of length $2^{2m}-1$ with two primitive nonzeros $\alpha $ and $\alpha ^{d}$ . The method used to obtain this result proves constraints on the number of roots that certain seventh degree polynomials can have on the unit circle of a finite field. The method also works when $m$ is odd, in which case the associated crosscorrelation and Walsh spectra have at most six distinct values.

Let $$q=2^m$$ . The projective general linear group $${\mathrm {PGL}}(2,q)$$ acts as a 3-transitive permutation group on the set of points of the projective line. The first objective of this paper is to prove that all linear codes over $${\mathrm {GF}}(2^h)$$ that are invariant under $${\mathrm {PGL}}(2,q)$$ are trivial codes: the repetition code, the whole space $${\mathrm {GF}}(2^h)^{2^m+1}$$ , and their dual codes. As an application of this result, the 2-ranks of the (0,1)-incidence matrices of all $$3-(q+1,k,\lambda )$$ designs that are invariant under $${\mathrm {PGL}}(2,q)$$ are determined. The second objective is to present two infinite families of cyclic codes over $${\mathrm {GF}}(2^m)$$ such that the set of the supports of all codewords of any fixed nonzero weight is invariant under $${\mathrm {PGL}}(2,q)$$ , therefore, the codewords of any nonzero weight support a 3-design. A code from the first family has parameters $$[q+1,q-3,4]_q$$ , where $$q=2^m$$ , and $$m\ge 4$$ is even. The exact number of the codewords of minimum weight is determined, and the codewords of minimum weight support a 3- $$(q+1,4,2)$$ design. A code from the second family has parameters $$[q+1,4,q-4]_q$$ , $$q=2^m$$ , $$m\ge 4$$ even, and the minimum weight codewords support a 3- $$(q +1,q-4,(q-4)(q-5)(q-6)/60)$$ design, whose complementary 3- $$(q +1, 5, 1)$$ design is isomorphic to the Witt spherical geometry with these parameters. A lower bound on the dimension of a linear code over $${\mathrm {GF}}(q)$$ that can support a 3- $$(q +1,q-4,(q-4)(q-5)(q-6)/60)$$ design is proved, and it is shown that the designs supported by the codewords of minimum weight in the codes from the second family of codes meet this bound.

With entanglement-assisted (EA) formalism, arbitrary classical linear codes are allowed to transform into EAQECCs by using pre-shared entanglement between the sender and the receiver. In this paper, based on classical cyclic MDS codes by exploiting pre-shared maximally entangled states, we construct two families of $q$-ary entanglement-assisted quantum MDS codes $[[\frac{q^{2}+1}{a},\frac{q^{2}+1}{a}-2(d-1)+c,d;c]]$, where q is a prime power in the form of $am+l$, and $a=(l^2+1)$ or $a=\frac{(l^2+1)}{5}$. We show that all of $q$-ary EAQMDS have minimum distance upper limit much larger than the known quantum MDS (QMDS) codes of the same length. Most of these $q$-ary EAQMDS codes are new in the sense that their parameters are not covered by the codes available in the literature., arXiv admin note: substantial text overlap with arXiv:1803.04168, arXiv:1912.12031

For any $$(1+u^2)$$ -constacyclic code C with arbitrary length N over $$\mathbb {F}_2[u]/\langle u^3\rangle $$ , we construct a new map $$\Phi $$ from $$(\mathbb {F}_2[u]/\langle u^3\rangle )^{N}$$ to $$(\mathbb {F}_2[v]/\langle v^2\rangle )^{2N}$$ , and derive $$\Phi (C)$$ by applying the map to each codeword of C. We find that C and its image $$\Phi (C)$$ have the same homogeneous distance, and the cyclic shift of any codeword of $$\Phi (C)$$ is still contained in $$\Phi (C)$$ over $$\mathbb {F}_2[v]/\langle v^2\rangle $$ , but $$\Phi (C)$$ is not a cyclic code in some cases. These properties help us to determine the lower bound and the upper bound for the homogeneous distance of C by discussing the same problem of $$\Phi (C)$$ . With these bounds, exact homogeneous distances of some $$(1+u^2)$$ -constacyclic codes are given. On the other hand, we consider the application of homogeneous distances in computing Hamming distances of a class of binary linear quasi-cyclic codes, and derive some optimal binary linear quasi-cyclic codes.

An $(r, \delta)$ -locally repairable code ( $(r, \delta)$ -LRC for short) was introduced by Prakash et al. [14] for tolerating multiple failed nodes in distributed storage systems, which was a generalization of the concept of $r$ -LRCs produced by Gopalan et al. [5] . An $(r, \delta)$ -LRC is said to be optimal if it achieves the Singleton-like bound. Recently, Chen et al. [2] generalized the construction of cyclic $r$ -LRCs proposed by Tamo et al. [19] , [20] and constructed several classes of optimal $(r, \delta)$ -LRCs of length $n$ for $n\, |\, (q-1)$ or $n\,|\, (q+1)$ , respectively in terms of a union of the set of zeros controlling the minimum distance and the set of zeros ensuring the locality. Following the work of [2] , [3] , this paper first characterizes $(r, \delta)$ -locality of a cyclic code via its zeros. Then we construct several classes of optimal cyclic $(r, \delta)$ -LRCs of length $n$ for $n\, |\, (q-1)$ or $n\,|\, (q+1)$ , respectively from the product of two sets of zeros. Our constructions include all optimal cyclic $(r,\delta)$ -LRCs proposed in [2] , [3] , and our method seems more convenient to obtain optimal cyclic $(r, \delta)$ -LRCs with flexible parameters. Moreover, many optimal cyclic $(r,\delta)$ -LRCs of length $n$ for $n\, |\, (q-1)$ or $n\,|\, (q+1)$ , respectively with $(r+\delta -1)\nmid n$ can be obtained from our method.

Let [Formula: see text], where [Formula: see text] is a power of a prime number [Formula: see text] and [Formula: see text]. A triple cyclic code of length [Formula: see text] over [Formula: see text] is a set that can be partitioned into three parts that any cyclic shift of the coordinates of the three parts leaves the code invariant. These codes can be viewed as [Formula: see text]-submodules of [Formula: see text]. In this paper, we study the generator polynomials and the minimum generating sets of this kind of codes. Some optimal or almost optimal linear codes are obtained from this family of codes. We present the relationship between the generators of triple cyclic codes and their duals. As a special class of triple cyclic codes, separable codes over [Formula: see text] are discussed briefly in the end.

In this paper, a set of generators (in a unique from) called the distinguished set of generators, of a cyclic code C of length n = 2 k (where k is a natural number) over Z 2 m is obtained. This set of generators is used to find the rank of the cyclic code C . It is proved that the rank of a cyclic code C of length n = 2 k over Z 2 m is equal to n − v , where v is the degree of a minimal degree polynomial in C . Then a description of all MHDR (maximum hamming distance with respect to rank) cyclic codes of length n = 2 k over Z 2 m is given. An example of the best cyclic codes over Z 8 of length 4 having largest minimum Hamming, Lee and Euclidean distances among all cyclic codes of the same rank is also given. Further, using an isomorphism between cyclic and negacyclic codes of odd length over finite chain rings given by Dinh, Lopez and Szabo, the above results are extended to cyclic and negacyclic codes of length p k over Z p m for an odd prime p .

We classify all the cyclic self-dual codes of length $$p^k$$ over the finite chain ring $$\mathcal R:=\mathbb Z_p[u]/\langle u^3 \rangle $$ , which is not a Galois ring, where p is a prime number and k is a positive integer. First, we find all the dual codes of cyclic codes over $${\mathcal R}$$ of length $$p^k$$ for every prime p. We then prove that if a cyclic code over $${\mathcal R}$$ of length $$p^k$$ is self-dual, then p should be equal to 2. Furthermore, we completely determine the generators of all the cyclic self-dual codes over $$\mathbb Z_2[u]/\langle u^3 \rangle $$ of length $$2^k$$ . Finally, we obtain a mass formula for counting cyclic self-dual codes over $$\mathbb Z_2[u]/\langle u^3 \rangle $$ of length $$2^k$$ .

Let R be a Galois ring of characteristic p a and cardinality p a m , where p is a prime and a , m are natural numbers. For a unit u and an element k ∈ R such that u 2 = 1 and u k = k , a generalized notion of a complement of an element e in R with respect to u and k has been given in this paper. Using this notion of complement, reversible complement cyclic codes with respect to u and k have been defined. Further, a necessary and sufficient condition for a reversible cyclic code of arbitrary length over R to be a reversible complement cyclic code with respect to u and k has been obtained. This generalization is significant in view of the fact that a reversible cyclic code of length n over R may be reversible complement with respect to one choice of u and k but may not be reversible complement with respect to some other choice of u and k . Further, reversible complement cyclic codes with respect to u and k (for different values of u and k ) over a special Galois ring, Z 4 , have been used to construct some DNA codes. It has been observed that codes over Z 4 in some cases give better DNA codes than DNA codes over fields and some rings of equal cardinality. A tradeoff between the two DNA properties namely avoiding secondary structure formation and constant G C − content has also been observed.

For any odd positive integer n, we express cyclic codes over $${\mathbb {Z}}_4$$ of length 4n in a new way. Based on the expression of each cyclic code $${\mathcal {C}}$$, we provide an efficient encoder and determine the type of $${\mathcal {C}}$$. In particular, we give an explicit representation and enumeration for all distinct self-dual cyclic codes over $${\mathbb {Z}}_4$$ of length 4n and correct a mistake in the paper “Concatenated structure of cyclic codes over $${\mathbb {Z}}_4$$ of length 4n” (Cao et al. in Appl Algebra Eng Commun Comput 10:279–302, 2016). In addition, we obtain 50 new self-dual cyclic codes over $${\mathbb {Z}}_4$$ of length 28.

In this paper, we study the algebraic structure of additive cyclic codes over the alphabet $${\mathbb {F}}_{2}^{r}\times {\mathbb {F}}_{4}^{s}={ \mathbb {F}}_{2}^{r}{\mathbb {F}}_{4}^{s},$$ where r and s are non-negative integers, $$\mathbb {F}_{2}={\mathbb {GF}}(2)$$ and $$\mathbb {F}_{4}={\mathbb {GF}} (4)$$ are the finite fields of 2 and 4 elements, respectively. We determine generator polynomials for $$\mathbb {F}_{2}\mathbb {F}_{4}$$ -additive cyclic codes. We also introduce a linear map W that depends on the trace map T to relate these codes to binary linear codes over $$\mathbb {F} _{2}.$$ Further, we investigate the duals of $$\mathbb {F}_{2}\mathbb {F}_{4}$$ -additive cyclic codes. We show that the dual of any $$\mathbb {F}_{2}\mathbb {F }_{4}$$ -additive cyclic code is another $$\mathbb {F}_{2}\mathbb {F}_{4}$$ -additive cyclic code. Using the mapping W, we provide examples of $$\mathbb {F}_{2}\mathbb {F}_{4}$$ -additive cyclic codes whose binary images have optimal parameters. We also consider additive cyclic codes over $$\mathbb {F}_{4}$$ and give some examples of optimal parameter quantum codes over $$\mathbb {F}_{4}$$ .

Upadhyay, Ashish Kumar/0000-0001-6307-6799 WOS: 000519450400019 In this paper, we study the quantum codes over , which are obtained from (lambda(1) + lambda(2))-constacyclic codes over the semi local ring , where v(2) = 1, p is odd prime. We decompose a (lambda(1) + lambda(2))-constacyclic code over into two constacyclic codes over such as (lambda(1) + lambda(2))-constacyclic and (lambda(1)-lambda(2))-constacyclic. We give the necessary and sufficient condition that the (lambda(1) + v lambda(2))-constacyclic codes over contain their duals. We give some examples of non binary quantum codes. University Grant Commission (UGC)University Grants Commission, India [2061441025, 22/06/2014(i)EU-V] The first author is thankful to University Grant Commission (UGC), Govt. of India for financial support under Sr. No. 2061441025 with Ref No. 22/06/2014(i)EU-V.

Motivated by the recent studies on $\mathbb {Z}_{2}\mathbb {Z}_{4}$ -additive cyclic codes, $\mathbb {Z}_{2}\mathbb {Z}_{2}[u]$ -cyclic codes and $\mathbb {Z}_{2}\mathbb {Z}_{2^{s}}$ -additive codes have been introduced by Aydogdu et al. . In this paper, we study $\mathbb {Z}_{q}R$ -linear cyclic codes where $R=\mathbb {Z}_{q}+u\mathbb {Z}_{q}, q$ is a prime number and $u^{2}=0$ . There are two major ingredients in this paper. The first is to investigate the algebraic structure of cyclic codes and their duals over ring $\mathbb {Z}_{q}R$ , the spanning sets, the types and sizes of $\mathbb {Z}_{q}R$ -cyclic codes and their duals as well. Based on this, the second ingredient is to present an infinite family of MDSS codes and obtain some illustrative examples of $q$ -ary cyclic codes with optimal parameters derived from the $\mathbb {Z}_{q}R$ -cyclic codes.

In this work, we investigate DNA codes of odd length over the finite ring R = Z 4 +vZ 4 , v 2 = v. Firstly, we give a map Φ from R n to {A, T, G, C} 2n . Then cyclic codes of odd length satisfying the reverse and reverse-complement constraint are characterized over such ring. Moreover, when C is a reverse-complement cyclic code over R, we prove that Φ(C) is a cyclic DNA code by using the method different from all the previously known ones, which helps us finding a new DNA code with 256 codewords. Finally, we consider the GC content of code C over finite ring Z 4 + vZ 4 by means of the Gray images of the minimal generating set of C.

In this paper, we study cyclic codes of length n over R = Zq + uZq, u2 = 0, where q is a power of a prime p and (n; p) = 1. We have determined the complete ideal structure of R. Using this, we have obtained the structure of cyclic codes and that of their duals through the factorization of xn-1 over R. We have also computed total number of cyclic codes of length n over R. A necessary and sufficient condition for a cyclic code over R to be self-dual is presented. We have presented a formula for the total number of self-dual cyclic codes of length n over R. A new Gray map from R to Z2rp is defined. Using Magma, some good cyclic codes of length 4 over Z9 + uZ9 are obtained.

In this paper, we develop the method for constructing DNA codes of odd length over the finite chain ring $R=\mathbb {F}_{4}[u]/\langle u^{3}\rangle $ , which plays an important role in genetics, bioengineering and DNA computing. By using a Gray map $\eta $ from $R$ to $\mathbb {F}_{4}$ , we give a one-to-one correspondence between 64 DNA codons of the alphabet $\{A, T, G,C\}^{3}$ and the 64 elements of the chain ring $R$ . We also establish a map $\Theta $ from $R^{n}$ to $\{A, T, G,C\}^{3n}$ . Then we provide a necessary and sufficient condition for cyclic codes of odd length over the chain ring $R$ to be reversible. Finally, when $C$ is a reversible cyclic code over $R$ , we derive a necessary and sufficient condition for $\Theta (C)$ to be a DNA code.

For a prime p and $$q=p^{m}$$ , we study reversible cyclic codes of arbitrary length over a ring $$ R = \mathbb {F}_q + u \mathbb {F}_q$$ where $$u^2=0 ~\mathrm{{mod}}~q$$ . First, we find a unique set of generators for cyclic codes over R followed by a classification of reversible cyclic codes concerning their generators. Further, we find the set of generators for dual codes, and then under certain conditions, it is shown that dual of reversible cyclic code over $$\mathbb {Z}_2+u\mathbb {Z}_2$$ is reversible. Moreover, we discuss reversible-complement cyclic codes over $$\mathbb {F}_8+u\mathbb {F}_8$$ , which play a very significant role in DNA computing. Finally, to show the importance of these results, examples of reversible cyclic codes and DNA codes are provided.

Let $$\mathbb {F}_{2^m}$$ be a finite field of $$2^m$$ elements and denote $$R=\mathbb {F}_{2^m}[u]/\langle u^k\rangle $$ $$=\mathbb {F}_{2^m}+u\mathbb {F}_{2^m}+\cdots +u^{k-1}\mathbb {F}_{2^m}$$ ( $$u^k=0$$ ), where k is an integer satisfying $$k\ge 2$$ . For any odd positive integer n, an explicit representation for every self-dual cyclic code over R of length 2n and a mass formula to count the number of these codes are given. In particular, a generator matrix is provided for the self-dual 2-quasi-cyclic code of length 4n over $$\mathbb {F}_{2^m}$$ derived by an arbitrary self-dual cyclic code of length 2n over $$\mathbb {F}_{2^m}+u\mathbb {F}_{2^m}$$ and a Gray map from $$\mathbb {F}_{2^m}+u\mathbb {F}_{2^m}$$ onto $$\mathbb {F}_{2^m}^2$$ . Finally, the hull of each cyclic code with length 2n over $$\mathbb {F}_{2^m}+u\mathbb {F}_{2^m}$$ is determined and all distinct self-orthogonal cyclic codes of length 2n over $$\mathbb {F}_{2^m}+u\mathbb {F}_{2^m}$$ are listed.

Let $R=\mathbb {Z}_{4}[v]/\langle v^{2}+2v\rangle $. Then R is a local non-principal ideal ring of 16 elements. First, we give the structure of every cyclic code of odd length n over R and obtain a complete classification for these codes. Then we determine the cardinality, the type and its dual code for each of these cyclic codes. Moreover, we determine all self-dual cyclic codes of odd length n over R and provide a clear formula to count the number of these self-dual cyclic codes. Finally, we list some optimal 2-quasi-cyclic self-dual linear codes of length 30 over $\mathbb {Z}_{4}$ and obtain 4-quasi-cyclic and formally self-dual binary linear [60,30,12] codes derived from cyclic codes of length 15 over $\mathbb {Z}_{4}[v]/\langle v^{2}+2v\rangle $.

For odd prime $p$, this paper studies $(1+(p-2)u)$-constacyclic codes over the ring $R= \mathbb{Z}_{p} [u,v]/\langle u^2-u,v^2-v,uv-vu\rangle$. We show that the Gray images of $(1+(p-2)u)$-constacyclic codes over $R$ are cyclic and permutation equivalent to a quasi cyclic code over $\mathbb{Z}_{p}$. We derive the generators for $(1+(p-2)u)$-constacyclic and principally generated $(1+(p-2)u)$-constacyclic codes over $R$. Among others, we extend our results for skew $(1+(p-2)u)$-constacyclic codes over $R$ and exhibit the relation between skew $(1+(p-2)u)$-constacyclic codes with the other linear codes. Finally, as an application of our study, we compute several non trivial linear codes by using the Gray images of $(1+(p-2)u)$-constacyclic codes over this ring $R$.

We explicitly determine generators of cyclic codes over a non-Galois finite chain ring Z p [ u ] / 〈 u 3 〉 of length p k , where p is a prime number and k is a positive integer. We completely classify that there are three types of principal ideals of Z p [ u ] / 〈 u 3 〉 and four types of non-principal ideals of Z p [ u ] / 〈 u 3 〉 , which are associated with cyclic codes over Z p [ u ] / 〈 u 3 〉 of length p k . We then obtain a mass formula for cyclic codes over Z p [ u ] / 〈 u 3 〉 of length p k .

We construct a class of $\mathbb {Z}_{p}\mathbb {Z}_{p^{s}}$-additive cyclic codes generated by pairs of polynomials, where p is a prime number. The generator matrix of this class of codes is obtained. By establishing the relationship between the random $\mathbb {Z}_{p}\mathbb {Z}_{p^{s}}$-additive cyclic code and random quasi-cyclic code of index 2 over $\mathbb {Z}_{p}$, the asymptotic properties of the rates and relative distances of this class of codes are studied. As a consequence, we prove that $\mathbb {Z}_{p}\mathbb {Z}_{p^{s}}$-additive cyclic codes are asymptotically good since the asymptotic GV-bound at $\frac {1+p^{s-1}}{2}\delta $ is greater than $\frac {1}{2}$, the relative distance of the code is convergent to δ, while the rate is convergent to $\frac {1}{1+p^{s-1}}$ for $0< \delta < \frac {1}{1+p^{s-1}}$.

Let F 2 m be a finite field of cardinality 2 m and s a positive integer. Using properties for Kronecker product of matrices and calculation for linear equations over F 2 m , an efficient method for the construction of all distinct self-dual cyclic codes with length 2 s over the finite chain ring F 2 m + u F 2 m ( u 2 = 0 ) is provided. On that basis, an explicit representation for every self-dual cyclic code of length 2 s over F 2 m + u F 2 m and an exact formula to count the number of all these self-dual cyclic codes are given.

Let t ≥ 2 be an integer, and let p1, ⋯, pt be distinct primes. By using algebraic properties, the present paper gives a sufficient and necessary condition for the existence of non-trivial self-orthogonal cyclic codes over the ring $${Z_{{p_1}{p_2} \cdots {p_t}}}$$ and the corresponding explicit enumerating formula. And it proves that there does not exist any self-dual cyclic code over $${Z_{{p_1}{p_2} \cdots {p_t}}}$$ .

Let F q be a finite field and n a positive integer. In this paper, we find a new combinatorial method to determine weight enumerators of reducible cyclic codes and their dual codes of length n over F q , which just generalize results of Zhu et al. (2015); especially, we also give the weight enumerator of a cyclic code, which is viewed as a partial Melas code. Furthermore, weight enumerators obtained in this paper are all in the form of power of a polynomial.

In this article, keeping the huge research prospective of the study in mind, we consider the non-commutative ring $${\mathcal {M}}_4({\mathbb {F}}_2)$$ , the set of all $$4 \times 4$$ matrices over the field $${\mathbb {F}}_2$$ and confirm that this ring is isomorphic with the ring $${\mathbb {F}}_{16}+u {\mathbb {F}}_{16}+u^2 {\mathbb {F}}_{16}+u^3{\mathbb {F}}_{16}$$ , where $$u^4=0$$ . Besides, we develop the structure of cyclic codes and their generators over the ring. Also, making use of Gray map from $${\mathcal {M}}_4({\mathbb {F}}_2)$$ to $${\mathbb {F}}_{16}^4$$ , we infer that the image of a cyclic code is a linear code. Finally, our findings are authenticated by suitable non-trivial examples.

A general criterion to obtain $s$ -PD-sets of minimum size $s+1$ for partial permutation decoding, which enable correction up to $s$ errors, for systematic codes over a finite field ${ {F}}_{q}$ and ${ {Z}}_{4}$ -linear codes is provided. We show how this technique can be easily applied to linear cyclic codes over ${ {F}}_{q}$ , ${ {Z}}_{4}$ -linear codes which are the Gray map image of a quaternary linear cyclic code, and some related codes such as quasi-cyclic codes. Furthermore, specific results for some linear and nonlinear binary codes, including simplex, Kerdock, Delsarte-Goethals, and extended dualized Kerdock codes are given. Finally, applying this technique, new $s$ -PD-sets of size $s+1$ for ${ {Z}}_{4}$ -linear Hadamard codes of type $2^\gamma 4^\delta $ , for all $\delta \geq 4$ and $1 ; and for ${ {Z}}_{4}$ -linear simplex codes of type $4^{m}$ , for all $m\geq 2$ and $1 , are also provided.

Symbol-pair codes were introduced by Cassuto and Blaum in 2010 to protect pair errors in symbol-pair read channels. Recently Yaakobi, Bruck and Siegel (2016) generalized this notion to b -symbol codes in order to consider consecutive b errors for a prescribed integer $\text {b} \ge 2$ , and they gave constructions and decoding algorithms. Cyclic codes were considered by various authors as candidates for symbol-pair codes and they established minimum distance bounds on (certain) cyclic codes. In this paper we use algebraic curves over finite fields in order to obtain tight lower and upper bounds on b -symbol Hamming weights of arbitrary cyclic codes over $\mathbb {F}_{\text {q}}$ . Here $\text {b} \ge 2$ is an arbitrary prescribed positive integer and $\mathbb {F}_{\text {q}}$ is an arbitrary finite field. We also present a stability theorem for an arbitrary cyclic code $C$ of dimension $k$ and length n : the b -symbol Hamming weight enumerator of C is the same as the k -symbol Hamming weight enumerator of C if $\text {k} \le \text {b} \le \text {n}-1$ . Moreover, we give improved tight lower and upper bounds on b -symbol Hamming weights of some cyclic codes related to irreducible cyclic codes. Throughout the paper the length n is coprime to q .

Let $$\mathrm {R}={\mathbb {F}}_4+v{\mathbb {F}}_4, v^2=v$$ . A linear code over $$\mathrm {R}$$ is a double cyclic code of length (r, s), if the set of its coordinates can be partitioned into two parts of sizes r and s, so that any cyclic shift of coordinates of both parts leave the code invariant. In polynomial representation, these codes can be viewed as $$\mathrm {R}[x]$$ -submodules of $$\frac{\mathrm {R}[x]}{\langle x^r-1\rangle }\times \frac{\mathrm {R}[x]}{\langle x^s-1\rangle }$$ . In this paper, we determine generator polynomials of $$\mathrm {R}$$ -double cyclic codes and their duals for arbitrary values of r and s. We enumerate $$\mathrm {R}$$ -double cyclic codes of length $$(2^{e_1},2^{e_2})$$ by giving a mass formula, where $$e_1$$ and $$e_2$$ are positive integers. Some structural properties of double constacyclic codes over $$\mathrm {R}$$ are also studied. These results are illustrated with some good examples.

Let R be the polynomial residue ring $${\mathbb {F}}_{q^{2}}+u{\mathbb {F}}_{q^{2}}$$ , where $${\mathbb {F}}_{q^2}$$ is the finite field with $$q^2$$ elements, q is a power of a prime p, and u is an indeterminate with $$u^{2}=0.$$ We introduce a Gray map from R to $${\mathbb {F}}_{q^{2}}^{p}$$ and study $$(1-u)$$-constacyclic codes over R. It is proved that the image of a $$(1-u)$$-constacyclic code of length n over R under the Gray map is a distance-invariant linear cyclic code of length pn over $${\mathbb {F}}_{q^{2}}.$$ We give some necessary and sufficient conditions for $$(1-u)$$-constacyclic codes over R to be Hermitian dual-containing. In particular, a new class of $$2^{m}$$-ary quantum codes is obtained via the Gray map and the Hermitian construction from Hermitian dual-containing $$(1-u)$$-constacyclic codes over the ring $${\mathbb {F}}_{2^{2m}}+u{\mathbb {F}}_{2^{2m}}$$.

We investigate the largest number of nonzero weights of quasi-cyclic codes. In particular, we focus on the function $\Gamma _{Q}(n,\ell,k,q)$ , that is defined to be the largest number of nonzero weights a quasi-cyclic code of index $\gcd (\ell,n)$ , length $n$ and dimension $k$ over $\mathbb F_{q}$ can have, and connect it to similar functions related to linear and cyclic codes. We provide several upper and lower bounds on this function, using different techniques and studying its asymptotic behavior. Moreover, we determine the smallest index for which a $q$ -ary Reed-Muller code is quasi-cyclic, a result of independent interest.

Upper and lower bounds on the largest number of weights in a cyclic code of given length, dimension and alphabet are given. An application to irreducible cyclic codes is considered. Sharper upper bounds are given for the special cyclic codes (called here strongly cyclic), {whose nonzero codewords have period equal to the length of the code}. Asymptotics are derived on the function $\Gamma(k,q),$ {that is defined as} the largest number of nonzero weights a cyclic code of dimension $k$ over $\F_q$ can have, and an algorithm to compute it is sketched. The nonzero weights in some infinite families of Reed-Muller codes, either binary or $q$-ary, as well as in the $q$-ary Hamming code are determined, two difficult results of independent interest., Comment: submitted on 21 June, 2018

In this article, for any odd prime p, we study the cyclic codes over the finite ring \(R=\mathbb {F}_{p}[v,w]/\langle v^{2}-1,w^{2}-1,vw-wv\rangle \) to obtain the quantum codes over \(\mathbb {F}_{p}\). We obtain the necessary and sufficient condition for cyclic codes which contain their duals and as an application, some new quantum codes are presented at the end of the article.

The projective special linear group ${\mathrm {PSL}}_{2}(n)$ is 2-transitive for all primes $n$ and 3-homogeneous for $n \equiv 3 \pmod {4}$ on the set $\{0,1, \ldots, n-1, \infty \}$ . It is known that the extended odd-like quadratic residue codes are invariant under ${\mathrm {PSL}}_{2}(n)$ . Hence, the extended quadratic residue codes hold an infinite family of 2-designs for primes $n \equiv 1 \pmod {4}$ , an infinite family of 3-designs for primes $n \equiv 3 \pmod {4}$ . To construct more $t$ -designs with $t \in \{2, 3\}$ , one would search for other extended cyclic codes over finite fields that are invariant under the action of ${\mathrm {PSL}}_{2}(n)$ . The objective of this paper is to prove that the extended quadratic residue binary codes are the only nontrivial extended binary cyclic codes that are invariant under ${\mathrm {PSL}}_{2}(n)$ .

Kavut, Selçuk (Balikesir Author), In 2003, Moreno and Castro proved that the covering radius of a class of primitive cyclic codes over the finite field F2 having minimum distance 5 (resp. 7) is 3 (resp. 5). We here give a generalization of this result as follows: the covering radius of a class of primitive cyclic codes over F2 with minimum distance greater than or equal to r+2 is r, where r is any odd integer. Moreover, we prove that the primitive binary e-error correcting BCH codes of length 2f-1 have covering radii 2e-1 for an improved lower bound of f., Gebze Teknik University - BAP 2015-A17

Let $$R = {{\mathbb {F}}_q} + {v_1}{{\mathbb {F}}_q} + \cdots + {v_r}{{\mathbb {F}}_q},$$ where q is a power of a prime, $$v_i^2=v_i,\; v_iv_j=v_jv_i=0$$ for $$1\le i,j \le r$$ and $$r\ge 1$$ . In this paper, the structure of cyclic codes over the ring R is studied and a Gray map $$\phi $$ from $${R^n}$$ to $${\mathbb {F}}_q^{(r + 1)n}$$ is given. We give a construction of quantum codes from cyclic codes over the ring R. We derive Euclidean dual containing codes over $${\mathbb {F}}_q$$ and Hermitian dual containing codes over $${\mathbb {F}}_{p^{2m}}$$ as Gray images of cyclic codes over R. In particular, we use $$r+1$$ codes associated with a cyclic code over R of arbitrary length to determine the parameters of the corresponding quantum code. Furthermore, some new non-binary quantum codes are obtained.

Let D 2 n = 〈 x , y ∣ x n = 1 , y 2 = 1 , y x y = x − 1 〉 be a dihedral group, and R = GR ( p 2 , m ) be a Galois ring of characteristic p 2 and cardinality p 2 m where p is a prime. Left ideals of the group ring R [ D 2 n ] are called left dihedral codes over R of length 2 n , and abbreviated as left D 2 n -codes over R . Let gcd ( n , p ) = 1 in this paper. Then any left D 2 n -code over R is uniquely decomposed into a direct sum of concatenated codes with inner codes A i and outer codes C i , where A i is a cyclic code over R of length n and C i is a skew cyclic code of length 2 over a Galois ring or principal ideal ring extension of R . Specifically, a generator matrix and basic parameters for each outer code C i are given. A formula to count the number of these codes is obtained and the dual code for each left D 2 n -code is determined. Moreover, all self-dual left D 2 n -codes and self-orthogonal left D 2 n -codes over R are presented.

In this paper, R is a finite chain ring with residue field F q and γ is a unit in R . By assuming that the multiplicative order u of γ is coprime to q , we give the trace-representation of any simple-root γ -constacyclic code over R of length l , and on the other hand show that any cyclic code over R of length u l is a direct sum of trace-representable cyclic codes. Finally, we characterize the simple-root, contractable and cyclic codes over R of length u l into γ -constacyclic codes of length l .

In this paper, we study cyclic codes over $$\mathbb {Z}_9$$ of length 3n, where n is a positive integer satisfying $$\mathrm{gcd}(3,n)=1$$ . First, a canonical form decomposition of any cyclic code over $$\mathbb {Z}_9$$ of length 3n are given and a unique set of generators for each subcode is presented. Hence the structure of any cyclic code over $$\mathbb {Z}_9$$ of length 3n is determined. From this decomposition, formulas for the number of all codes and the number of codewords in each code are given. Then dual codes and self-duality of these codes are investigated. As an application, all 10061824 distinct cyclic codes over $$\mathbb {Z}_9$$ of length 24 and all 544 self-dual codes among them are listed explicitly. Moreover, 280 new and good self-dual cyclic codes over $$\mathbb {Z}_9$$ with basic parameters $$\left( 24, 3^{24}, 3\right) $$ are obtained.

The generators for $$(1+\lambda u^2)$$ -constacyclic codes of arbitrary length over $$F_{p^m}[u]/\langle u^3\rangle $$ are given. As a special case, we get generators for $$(1+\lambda u^2)$$ -constacyclic codes with length $$p^e$$ over $$F_{p^m}[u]/\langle u^3\rangle $$ . Besides, we determine generators for the dual codes of $$(1+\lambda u^2)$$ -constacyclic codes with an arbitrary length over $$F_{p^m}[u]/\langle u^3\rangle $$ . Based on this, the necessary and sufficient condition for being a self-dual $$(1+\lambda u^2)$$ -constacyclic code over $$F_{2^m}[u]/\langle u^3\rangle $$ is derived.

Relative generalized Hamming weights (RGHWs) of a linear code respect to a linear subcode determine the security of the linear ramp secret sharing scheme based on the code. They can be used to express the information leakage of the secret when some keepers of shares are corrupted. Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems. In this paper, we investigate the RGHWs of cyclic codes with two nonzeros respect to any of its irreducible cyclic subcodes. Applying the method in the paper [arxiv.org/abs/1410.2702], we give two formulae for RGHWs of the cyclic codes. As applications of the formulae, explicit examples are computed., Comment: 14 pages

Let m and k be two positive integers such that v 2 ( m ) = v 2 ( k ) , where v 2 ( ⋅ ) denotes the 2-valuation function, and let α be a primitive element of F p 2 m . Let C be a cyclic code over F p with a parity-check polynomial h 1 ( x ) h 2 ( x ) h 3 ( x ) ∈ F p 2 m [ x ] , where h 1 ( x ) , h 2 ( x ) and h 3 ( x ) are the minimal polynomials of α − p k + 1 2 , − α − p k + 1 2 and α − p m + 1 2 over F p respectively. In this paper, we determine the weight distribution and the complete weight distribution of the code C .

Minimum weight bases of some extended cyclic codes can be chosen from the affine orbits of certain explicitly represented minimum weight codewords. We find such bases for the following three classes of codes: the extended primitive 2-error correcting BCH code of length \(n=2^m,\) where \(m\ge 4\) (for \(m\ge 20\) the result was proven in Grigorescu and Kaufman IEEE Trans Inf Theory 58(I. 2):78–81, 2011), the extended cyclic code \(\bar{C}_{1,5}\) of length \(n=2^m,\) odd m, \(m\ge 5,\) and the extended cyclic codes \(\bar{C}_{1,2^i+1}\) of lengths \(n=2^m,\) \((i,\,m)=1\) and \(3\le i\le \frac{m-5}{4}-o(m).\)

The aim of this paper is to establish all self-dual -constacyclic codes of length ps over the finite commutative chain ring R=Fpm+uFpm, where p is a prime and u2=0. If =+u for nonzero elements , of Fpm, the ideal u is the unique self-dual (+u)-constacyclic codes. If = for some nonzero element of Fpm, we consider two cases of . When =1, i.e., =1 or 1, we first obtain the dual of every cyclic code, a formula for the number of those cyclic codes and identify all self-dual cyclic codes. Then we use the ring isomorphism to carry over the results about cyclic accordingly to negacyclic codes. When 1, it is shown that u is the unique self-dual -constacyclic code. Among other results, the number of each type of self-dual constacyclic code is obtained.

In this paper, we study a class of skew-cyclic codes using a skew polynomial ring over \begin{document}$R = \mathbb{Z}_4+u\mathbb{Z}_4;u^2 = 1$ \end{document} , with an automorphism \begin{document}$θ$ \end{document} and a derivation \begin{document}$δ_θ$ \end{document} . We generalize the notion of cyclic codes to skew-cyclic codes with derivation, and call such codes as \begin{document}$δ_θ$ \end{document} -cyclic codes. Some properties of skew polynomial ring \begin{document}$R[x, θ, {δ_θ}]$ \end{document} are presented. A \begin{document}$δ_θ$ \end{document} -cyclic code is proved to be a left \begin{document}$R[x, θ, {δ_θ}]$ \end{document} -submodule of \begin{document}$\frac{R[x, θ, {δ_θ}]}{\langle x^n-1 \rangle}$ \end{document} . The form of a parity-check matrix of a free \begin{document}$δ_θ$ \end{document} -cyclic codes of even length \begin{document}$n$ \end{document} is presented. These codes are further generalized to double \begin{document}$δ_θ$ \end{document} -cyclic codes over \begin{document}$R$ \end{document} . We have obtained some new good codes over \begin{document}$\mathbb{Z}_4$ \end{document} via Gray images and residue codes of these codes. The new codes obtained have been reported and added to the database of \begin{document}$\mathbb{Z}_4$ \end{document} -codes [ 2 ].