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BCH codes have been widely used in consumer devices, communication systems, and data storage systems. In this paper, we study the dimensions of four families of the BCH codes C (q , n , δ , 0) of length n = q m + 1 , where designed distances δ are given as follows. (1) δ = a q m + 1 q + 1 + 1 , where 1 ≤ a ≤ ⌊ q − 1 2 ⌋ and m is odd. (2) δ = a q m − 1 q − 1 +2, where 1 ≤ a ≤ ⌊ q − 1 2 ⌋. (3) a q m − 1 q − 1 + 2 < δ < a q m − 1 q − 1 + (q − 2 a − 1) q m − 1 2 + 3 , where m is odd and 0 ≤ a ≤ ⌊ q − 3 2 ⌋. (4) a q m − 1 q − 1 + 2 < δ < a q m − 1 q − 1 + q m 2 + 4 , where m is even and 0 ≤ a ≤ ⌊ q − 3 2 ⌋. The minimum distance of the first family of the BCH codes C (q , n , δ , 0) is determined. The Bose distances of the third and fourth families of the BCH codes C (q , n , δ , 0) are determined under some conditions. Many optimal linear codes are obtained from these BCH codes. [ABSTRACT FROM AUTHOR]

Abstract 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. [ABSTRACT FROM AUTHOR]

Recently, Liu and Liu gave the Singleton bound for pure asymmetric entanglement-assisted quantum error-correcting (AEAQEC) codes. They constructed three new families of AQEAEC codes by means of Vandermonde matrices, generalized Reed-Solomon (GRS) codes and cyclic codes. In this paper, we first exhibit the Singleton bound for any AEAQEC codes. Then we construct AEAQEC codes by two distinct constacyclic codes. By means of repeated-root cyclic codes, we construct new AEAQEC MDS codes. In addition, our methods allow for easily calculating the dimensions, d z , d x and the number c of pre-shared maximally entangled states of AEAQEC codes. [ABSTRACT FROM AUTHOR]

Linear complementary dual (LCD) codes and linear complementary pairs (LCP) of codes have been proposed for new applications as countermeasures against side-channel attacks (SCA) and fault injection attacks (FIA) in the context of direct sum masking (DSM). The countermeasure against FIA may lead to a vulnerability for SCA when the whole algorithm needs to be masked (in environments like smart cards). This led to a variant of the LCD and LCP problems, where several results were obtained intensively for LCD codes, but only partial results were derived for LCP codes. Given the gap between the thin results and their particular importance, this paper aims to reduce this by further studying the LCP of codes in special code families and, precisely, the characterization and construction mechanism of LCP codes of algebraic geometry codes over finite fields. Notably, we propose constructing explicit LCP of codes from elliptic curves. Besides, we also study the security parameters of the derived LCP of codes (C , D) (notably for cyclic codes), which are given by the minimum distances d (C) and d (D ⊥). Further, we show that for LCP algebraic geometry codes (C , D) , the dual code C ⊥ is equivalent to D under some specific conditions we exhibit. Finally, we investigate whether MDS LCP of algebraic geometry codes exist (MDS codes are among the most important in coding theory due to their theoretical significance and practical interests). Construction schemes for obtaining LCD codes from any algebraic curve were given in 2018 by Mesnager, Tang and Qi in [11]. To our knowledge, it is the first time LCP of algebraic geometry codes has been studied. [ABSTRACT FROM AUTHOR]

BCH codes are a special subclass of cyclic codes. In many cases, BCH codes are best cyclic codes and they have wide applications in communication and storage systems. In this paper, we investigate the parameters of a class of narrow-sense BCH codes over G F (q) of length 2 (q 2 m − 1) q + 1 with small and large dimensions. We study the q -cyclotomic cosets modulo 2 (q 2 m − 1) q + 1 , determine the dimensions of these BCH codes and give the lower bounds on their minimum distances. Furthermore, we present the lower bounds on the minimum distances of their dual codes. [ABSTRACT FROM AUTHOR]

Constant dimension codes, a specialized subclass of subspace codes, have garnered significant attention in recent years due to their applications in random network coding. Among these codes, large cyclic constant dimension codes, endowed with optimal minimum distance, offer efficient solutions for encoding and decoding algorithms. This paper is dedicated to the design of cyclic constant dimension codes that achieve the dual goals: maximizing code size while maintaining optimal minimum distance. We first explore a new form of Sidon spaces and employ this new form to construct several families of cyclic subspace codes. Our codes have optimal minimum distances and have more codewords than previous constructions in the literature. [ABSTRACT FROM AUTHOR]

Let F p m be a finite field of cardinality p m and R 3 = F p m [ u ] 〈 u 3 〉 , where p is a prime, m is a positive integer and u 3 = 0. Laaouine et al. (2021) [7] obtained the torsion code and the number of codewords of cyclic codes of length p s over F p m [ u ] 〈 u 3 〉 , but some of the results were wrong, which in this paper, we improve them. By a new method, we compute the number of codewords of these codes. Finally, by using a one-to-one correspondence between cyclic and α -constacyclic codes of length p s over F p m [ u ] 〈 u 3 〉 , we generalize these results to α -constacyclic codes, where α is a non-zero element of the finite field F p m . [ABSTRACT FROM AUTHOR]

A locally repairable code (LRC) with locality r allows for the recovery of any erased codeword symbol using at most r other codeword symbols. The Singleton-type bound dictates a best possible tradeoff between the dimension, the minimum distance and the locality of LRCs. An LRC attaining this tradeoff is said to be optimal. A constacyclic LRC is an LRC with constacyclic structure which can be encoded efficiently using any encoding algorithm for cyclic codes in general. In this paper we consider the optimal constacyclic LRCs of length η p s over finite fields, where p is the characteristic of the finite fields, s is a positive integer and η is a positive integer coprime to p. We obtain several infinite classes of optimal constacyclic LRCs with new parameters. Furthermore, under a further assumption that the ambient space F p m [ x ] / 〈 x η p s − λ 〉 of the repeated-root constacyclic codes is a chain ring, we characterize all the possible optimal constacyclic LRCs in F p m [ x ] / 〈 x η p s − λ 〉. [ABSTRACT FROM AUTHOR]

Based on good algebraic structures and practicabilities, generalized quasi-cyclic (GQC) codes play important roles in coding theory. In this paper, we study some results on GQC codes over Z 4 including normalized generating sets, minimal generating sets and normalized generating sets of their dual codes. As an application, new Z 4 -linear codes and good binary nonlinear codes are constructed from GQC codes over Z 4. [ABSTRACT FROM AUTHOR]

In this paper, we study hulls of cyclic codes of length 2 n (i.e., oddly even length) over the ring Z 4 of integers modulo 4 by viewing these codes as ideals of the quotient ring Z 4 [ x ] / < x 2 n − 1 > , where n is an odd positive integer. We express the generator polynomials of the hull of each cyclic code C of length 2 n over Z 4 in terms of the generator polynomials of the code C , and we note that the 2-dimension of the hull of the code C is an integer Θ satisfying 0 ≤ Θ ≤ 2 n. We also obtain an enumeration formula for cyclic codes of length 2 n over Z 4 with hulls of a given 2-dimension. Besides this, we study the average 2-dimension, denoted by E (2 n) , of the hulls of cyclic codes of length 2 n over Z 4 and establish a formula for E (2 n) that handles well. With the help of this formula, we deduce that E (2 n) = 5 n 6 when n ∈ N 2 , and we study the growth rate of E (2 n) with respect to the length 2 n when n ∉ N 2 , where N 2 denotes the set of all positive integers ω such that w divides 2 i + 1 for some positive integer i. Further, when n ∉ N 2 , we derive lower and upper bounds on E (2 n) and show that these bounds are attained at n = 7. We also illustrate these results with some examples. As an application of these results, we construct some entanglement-assisted quantum error-correcting codes (EAQECCs) over Z 4 with specific parameters from cyclic codes of oddly even length over Z 4. [ABSTRACT FROM AUTHOR]

Let p ≠ 3 be any prime. The algebraic structures of all λ -constacyclic codes of length 3 p s over the finite commutative chain ring R = F p m + u F p m are provided by classifying all ideals of the local ring R [ x ] 〈 x 3 p s − λ 〉. Using these structures, in this paper, the exact values of Hamming distances of all such λ -constacyclic codes of length 3 p s are established. As an application, we identify all maximal distance separable λ -constacyclic codes of length 3 p s. [ABSTRACT FROM AUTHOR]

Wilson (1972) proved that for fixed positive integers d and r such that 2 d divides r (r − 1) , there exists a subset Y of size r of the finite field of order q such that the r (r − 1) nonzero differences occurring from Y are evenly distributed to the d th cyclotomic classes provided that q is sufficiently large relative to d and r. On the other hand, Kajiura, Matsumoto and Okuda (2019) introduced the concept of difference sets in commutative association schemes in their study on a quasi-Monte Carlo method to approximate integrations over the point sets. Then, Wilson's result implies the existence of difference sets in cyclotomic schemes. In this paper, we generalize Wilson's result into pseudocyclic commutative association schemes and show the existence of difference sets in pseudocyclic association schemes with sufficiently large numbers of points. [ABSTRACT FROM AUTHOR]

If a linear code is contained in its dual, then it is said to be self-orthogonal. Self-orthogonal codes are an important subclass of linear codes which have many nice applications. In this paper, we mainly study the augmented codes of the linear codes constructed by Hu et al. (2022) [21] , and the subcodes of the codes punctured from the irreducible cyclic codes constructed by Ding and Yang (2013) [11]. The parameters and weight distributions of these codes are explicitly determined. These codes are proved to be self-orthogonal under certain conditions. As an application, these self-orthogonal codes are used to derive optimal or almost optimal quantum codes with new parameters. [ABSTRACT FROM AUTHOR]

Abstract: Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have very efficient encoding and decoding algorithms. In this paper, monomials and trinomials over finite fields with even characteristic are employed to construct a number of families of binary cyclic codes. Lower bounds on the minimum weight of some families of the cyclic codes are developed. The minimum weights of other families of the codes constructed in this paper are determined. The dimensions of the codes are flexible. Some of the codes presented in this paper are optimal or almost optimal in the sense that they meet some bounds on linear codes. Open problems regarding binary cyclic codes from monomials and trinomials are also presented. [Copyright &y& Elsevier]

Abstract Let F 2 m be a finite field of cardinality 2 m , n be an odd positive integer, and denote R = F 2 m [ u ] ∕ 〈 u 3 〉. Let δ , α ∈ F 2 m ×. Then (δ + α u 2) -constacyclic codes over R are called constacyclic codes over R of Type 2. In this paper, an explicit representation and a complete description for all distinct (δ + α u 2) -constacyclic codes over R of length 2 n and their dual codes are given. Moreover, explicit formulas for the number of codewords in each code and the number of all such codes are provided respectively. In particular, all distinct self-dual (1 + α u 2) -constacyclic codes over R of length 2 n are presented precisely. In addition, a complement to a result in Cao et al. (2017) is given. [ABSTRACT FROM AUTHOR]

Dinh et al. (2018) [10] obtained all self-dual constacyclic codes of length p s over R 2 = F p m + u F p m , where p is a prime number and u 2 = 0. In this paper, we determine the structure of (Euclidean) dual of some special skew cyclic codes of length p s over R 2 , and establish all of them which are self-dual. As a special case, we conclude all of results appeared in the above paper for cyclic codes. [ABSTRACT FROM AUTHOR]

BCH codes are an important class of cyclic codes, and have wide applications in communication and storage systems. However, it is difficult to determine the parameters of BCH codes and only a few cases are known. In this paper, we mainly study three classes of BCH codes with n = q m − 1 , q 2 s − 1 q + 1 , q m − 1 q − 1. On one hand, we provide the parameters of C (n , q , δ , 1) and its dual code in some cases. On the other hand, we provide the sufficient and necessary conditions to guarantee that C (n , q , δ , 2) is a dually-BCH code. [ABSTRACT FROM AUTHOR]

In this paper, we study skew frame starters, which are strong frame starters that satisfy an additional "skew" property. We prove three new non-existence results for cyclic skew frame starters of certain types. We also construct several small examples of previously unknown cyclic skew frame starters by computer. [ABSTRACT FROM AUTHOR]

Separable codes can be used to construct fingerprints resistant to the averaging collusion attack on multimedia contents. In this paper, we first show an upper bound on the size of a 2 ‾ -separable code with length 5 by using known results. Then we propose an equivalent condition of such a code. Finally, by using cyclic difference matrices and orthogonal arrays, q -ary 2 ‾ -separable codes of length 5 are constructed for any positive integer q with gcd ⁡ (q , 12) = 1. [ABSTRACT FROM AUTHOR]

In this paper, we consider cyclic codes of odd length n over the local, non-chain ring R = Z 2 s [ u ] ∕ 〈 u k 〉 = Z 2 s + u Z 2 s + … + u k − 1 Z 2 s ( u k = 0 ) , for any integers s ≥ 1 and k ≥ 2 . An explicit algebraic representation of such codes is obtained. This algebraic structure is then used to establish the duals of all cyclic codes. Among others, all self-dual cyclic codes of odd length n over the ring R are determined. Moreover, some examples are provided which produce several optimal codes. [ABSTRACT FROM AUTHOR]

Additive cyclic codes over Galois rings were investigated in Cao et al. (2015). In this paper, we investigate the same problem but over a more general ring family, finite commutative chain rings. When we focus on non-Galois finite commutative chain rings, we observe two different kinds of additivity. One of them is a natural generalization of the study in Cao et al. (2015), whereas the other one has some unusual properties especially while constructing dual codes. We interpret the reasons of such properties and illustrate our results giving concrete examples. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 . [ABSTRACT FROM AUTHOR]

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 . [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish all self-dual λ -constacyclic codes of length p s over the finite commutative chain ring R = F p m + u F p m , where p is a prime and u 2 = 0 . If λ = α + u β for nonzero elements α , β of F p m , the ideal 〈 u 〉 is the unique self-dual ( α + u β ) -constacyclic codes. If λ = γ for some nonzero element γ of F p m , we consider two cases of γ . When γ = γ − 1 , i.e., γ = 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 φ to carry over the results about cyclic accordingly to negacyclic codes. When γ ≠ γ − 1 , it is shown that 〈 u 〉 is the unique self-dual γ -constacyclic code. Among other results, the number of each type of self-dual constacyclic code is obtained. [ABSTRACT FROM AUTHOR]

Binary linear codes with good parameters have important applications in secret sharing schemes, authentication codes, association schemes, and consumer electronics and communications. In this paper, we construct several classes of binary linear codes from vectorial Boolean functions and determine their parameters, by further studying a generic construction developed by Ding et al. recently. First, by employing perfect nonlinear functions and almost bent functions, we obtain several classes of six-weight linear codes which contain the all-one codeword, and determine their weight distribution. Second, we investigate a subcode of any linear code mentioned above and consider its parameters. When the vectorial Boolean function is a perfect nonlinear function or a Gold function in odd dimension, we can completely determine the weight distribution of this subcode. Besides, our linear codes have larger dimensions than the ones by Ding et al.’s generic construction. [ABSTRACT FROM AUTHOR]

Symbol-pair codes are proposed to guard against pair-errors in symbol-pair read channels. The minimum symbol-pair distance is of significance in determining the error-correcting capability of a symbol-pair code. One of the central themes in symbol-pair coding theory is the constructions of symbol-pair codes with the largest possible minimum symbol-pair distance. Maximum distance separable (MDS) and almost maximum distance separable (AMDS) symbol-pair codes are optimal and sub-optimal regarding the Singleton bound, respectively. In this paper, six new classes of AMDS symbol-pair codes are explicitly constructed through repeated-root cyclic codes. Remarkably, one class of such codes has unbounded lengths and the minimum symbol-pair distance of another class can reach 13. [ABSTRACT FROM AUTHOR]

In the paper, we firstly study the algebraic structures of Z p Z p k -additive cyclic codes and give the generator polynomials and the minimal spanning set of these codes. Secondly, a necessary and sufficient condition for the Z p Z p 2 -additive cyclic code whose Gray image is linear (not necessarily cyclic) over Z p is given. Moreover, as for some special families of cyclic codes over Z p 2 and Z p Z p 2 , the linearity of the Gray images is determined. [ABSTRACT FROM AUTHOR]

In this paper, we explicitly determine the generator polynomials of all repeated-root constacyclic codes of length n l p s over F q and their dual codes, where l is an odd prime different from p , and n is an odd prime different from both l and p such that n = 2 h + 1 for some prime h or h = 1 . Moreover, we give all self-dual cyclic codes of length n l p s over F q and their enumeration. All linear complementary dual (LCD) cyclic codes of length n l p s over F q and their enumeration are also obtained. [ABSTRACT FROM AUTHOR]

Bierbrauer (2012) developed the theory of q -linear cyclic codes over ( F q ) m and he obtained a parametric description of such codes by cyclotomic cosets. Recently, Cao et al. (2015) obtained the structure of cyclic additive codes over the Galois ring GR ( p ℓ , m ) , where m is a prime integer. Let R be a finite commutative ring and R n = R [ x ] ∕ 〈 x n − 1 〉 . In this paper, we generalize the theory of F q -linear codes over vector spaces to R -linear codes over free R -algebras (free as R -module). We call these codes, R -additive codes. We introduce a one-to-one correspondence between the classes of cyclic R -additive code and the classes of R n -linear code. Using the structure of R n -linear codes, we present the structure of cyclic R -additive codes, where R is a chain ring. Among other results, q -linear cyclic codes over ( F q ) m are described by ring-theoretic facts, and the structure of cyclic additive codes over the Galois ring GR ( p ℓ , m ) is given for an arbitrary integer m , not necessarily a prime number. [ABSTRACT FROM AUTHOR]

Let F q be a finite field of odd order q and n = 2 a p 1 a 1 p 2 a 2 , where a , a 1 , a 2 are positive integers, p 1 , p 2 are distinct odd primes and 4 p 1 p 2 | q − 1 . In this paper, we study the irreducible factorization of x n − 1 over F q and all primitive idempotents in the ring F q [ x ] ∕ 〈 x n − 1 〉 .Moreover, we obtain the dimensions and the minimum Hamming distances of all irreducible cyclic codes of length n over F q . [ABSTRACT FROM AUTHOR]

In this paper, we first generalize the l -linear intersection pair (l -LIP) of codes over finite fields to the linear l -intersection pair of codes over finite chain rings. Then three criteria of l -linear intersection pair of codes over finite chain rings are obtained. By means of the generator set of cyclic codes, we obtain the generator set of the intersection of two cyclic codes which helps us to get a characterization of l -cyclic intersection pair of codes. As an application, l -cyclic intersection pair of codes over finite chain rings are used to construct new entanglement-assisted quantum error correcting codes. Finally, under suitable conditions, we also obtain a characterization for a l -cyclic intersection pair of codes over principal ideal rings Z k. [ABSTRACT FROM AUTHOR]

Finding the weight distributions of specific cyclic codes has been an interesting area of research. Although the class of cyclic codes has been studied for many years, their weight distributions are known only for a few cases. In this paper, we use matrix method to determine the weight distribution of a family of cyclic codes. [ABSTRACT FROM AUTHOR]

Cyclic codes have been an important topic of both mathematics and engineering for decades. They have been widely used in consumer electronics, data transmission technologies, broadcast systems, and computer applications as they have efficient encoding and decoding algorithms. The objective of this paper is to provide a survey of three-weight cyclic codes and their weight distributions. Information about the duals of these codes is also given when it is available. [ABSTRACT FROM AUTHOR]

Let F r be a finite field with r = q m elements and θ a primitive element of F r . Suppose that h 1 ( x ) and h 2 ( x ) are the minimal polynomials over F q of g 1 − 1 and g 2 − 1 , respectively, where g 1 , g 2 ∈ F r ∗ . Let C be a reducible cyclic code over F q with check polynomial h 1 ( x ) h 2 ( x ) . In this paper, we investigate the complete weight enumerators of the cyclic codes C in the following two cases: (1) g 1 = θ q − 1 h , g 2 = β g 1 , where h > 1 is a divisor of q − 1 , e > 1 is a divisor of h , and β = θ r − 1 e ; (2) g 1 = θ 2 , g 2 = θ p k + 1 , where q = p is an odd prime and k is a positive integer. Moreover, we explicitly present the complete weight enumerators of some cyclic codes. [ABSTRACT FROM AUTHOR]

Let p be an odd prime, and m and k be two positive integers with m ≥ 3 . Let h ± 1 ( x ) and h ± t ( x ) be the minimal polynomials of ± α − 1 and ± α − t over F p , respectively, where α is a primitive element of F p m . Let C 1 , − 1 , ± t , C ± 1 , t , − t and C 1 , − 1 , t , − t be the cyclic codes over F p of length p m − 1 with parity-check polynomials h 1 ( x ) h − 1 ( x ) h ± t ( x ) , h ± 1 ( x ) h t ( x ) h − t ( x ) and h 1 ( x ) h − 1 ( x ) h t ( x ) h − t ( x ) , respectively. This paper determines the weight distributions of the cyclic codes C 1 , − 1 , ± t , C ± 1 , t , − t and C 1 , − 1 , t , − t for the parameter t satisfying some congruence equations. [ABSTRACT FROM AUTHOR]

Let F q be a finite field with q elements, l an odd prime, and t , v positive integers such that l v | ( q − 1 ) and gcd ( t , l ) = 1 . In this paper, we give a combinatorial result and use it to determine the weight distribution of a cyclic code of length t l m with t | ( q − 1 ) , which is an open question in Yang et al. (2013). Moreover we compute the weight distributions of cyclic codes of length t l m , where q ≡ 3 ( mod 4 ) and t = 4 or 8 . [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the explicit expressions for the generating idempotents, check polynomials and the parameters of all minimal cyclic codes of length over are obtained, where is an odd prime different from the characteristic of , and are positive integers with , and . Our results generalize the main results in Pruthi and Arora (1997) and Arora and Pruthi (1999), which considered the cases and respectively. We propose an approach different from those in Pruthi and Arora (1997) and Arora and Pruthi (1999) to obtain the generating idempotents. [Copyright &y& Elsevier]

Let R be a finite commutative chain ring, D 2 n be the dihedral group of size 2 n and R [ D 2 n ] be the dihedral group ring. In this paper, we completely characterize left ideals of R [ D 2 n ] (called left D 2 n -codes) when gcd (c h a r (R) , n) = 1. In this way, we explore the structure of some skew-cyclic codes of length 2 over R and also over R × S , where S is an isomorphic copy of R. As a particular result, we give the structure of cyclic codes of length 2 over R. In the case where R = F p m is a Galois field, we give a classification for left D 2 N -codes over F p m , for any positive integer N. In both cases we determine dual codes and identify self-dual ones. [ABSTRACT FROM AUTHOR]

Abstract: The objectives of this paper are to survey and extend earlier results on the weight distributions of irreducible cyclic codes, present a divisibility theorem and develop bounds on the weights in irreducible cyclic codes. [Copyright &y& Elsevier]

Abstract: Cyclic codes of length over are precisely the ideals of the local ring . In this paper, we investigate the distances of cyclic codes of length over . The Hamming distances of all cyclic codes of length over are determined. The exact Lee distances of some cyclic codes of length over are also obtained. [Copyright &y& Elsevier]

In this paper we study the uncertainty principle (UP) connecting a function over a finite field and its Mattson-Solomon polynomial, which is a kind of Fourier transform in positive characteristic. Three versions of the UP over finite fields are studied, in connection with the asymptotic theory of cyclic codes. We first show that no finite field satisfies the strong version of UP, introduced recently by Evra, Kowalsky, Lubotzky, 2017. A refinement of the weak version is given, by using the asymptotic Plotkin bound. A naive version, which is the direct analogue over finite fields of the Donoho-Stark bound over the complex numbers, is proved by using the BCH bound. It is strong enough to show that there exist sequences of cyclic codes of length n , arbitrary rate, and minimum distance Ω (n α) for all 0 < α < 1 / 2. Finally, a connection with Ramsey Theory is pointed out. [ABSTRACT FROM AUTHOR]

Let F q be a finite field of order q , a prime power integer, such that q = e t + 1 where t ≥ 1 , e ≥ 2 are integers. In this paper, we study cyclic codes of length n over a non-chain ring R e , q = F q [ u ] / 〈 u e − 1 〉. We define a Gray map φ and obtain many maximum-distance-separable (MDS) and optimal F q -linear codes from the Gray images of cyclic codes. Under certain conditions we determine linear complementary dual (LCD) codes of length n when gcd ⁡ (n , q) ≠ 1 and gcd ⁡ (n , q) = 1 , respectively. It is proved that a cyclic code C of length n is an LCD code if and only if its Gray image φ (C) is an LCD code of length en over F q. Among others, we present the conditions for existence of free and non-free LCD codes. Moreover, we obtain many optimal LCD codes as the Gray images of non-free LCD codes over R e , q. [ABSTRACT FROM AUTHOR]

Let n be a positive integer with a factor k such that n ≥ 3 k. Let q be a prime power, and let G q (n , k) be the set of all k -dimensional F q -subspaces of the field F q n . In this paper, we construct cyclic subspace codes in G q (n , k) with minimum distance 2 k − 2 and size (⌈ n 2 k ⌉ − 1) ⋅ (q n − 1) q k q − 1 . In the case n = 3 k , their sizes differ from the sphere-packing bound for subspace codes by a factor of 1 q − 1 asymptotically as k goes to infinity. Our construction makes use of variants of the Sidon spaces constructed by Roth et al. (2018) and analogous to the results they attained for the case n = 2 k. We also establish the existence of Sidon spaces of G q (7 k , 2 k) , and thus we resolve part of the conjecture about the existence of cyclic subspace codes in G q (n , k) with minimum distance 2 k − 2 and size q n − 1 q − 1. [ABSTRACT FROM AUTHOR]

Let p be an odd prime, and k an integer such that k ∣ (p − 1). This paper studies quantum codes from skew cyclic codes over the ring R = F q [ u ] 〈 u k + 1 − u 〉 , where q is a power of the prime p. We construct a set of orthogonal idempotents of the ring R and using the set, skew cyclic codes over the ring R are decomposed as direct sum of skew cyclic codes over F q. We obtain a necessary and sufficient condition for a skew cyclic code to contain its dual. As an application, we construct quantum codes from skew cyclic codes over F q. It is observed that some quantum codes we constructed are MDS quantum codes. [ABSTRACT FROM AUTHOR]

A weak order is a way to rank n objects where ties are allowed. In this paper, we develop a generic framework to generate Gray codes for weak orders. We then describe a simple algorithm based on the framework that generates cyclic 2-Gray codes for weak orders in constant amortized time per string. This is the first known cyclic 2-Gray codes for weak orders. The framework can easily be modified to generate other Gray codes for weak orders. We provide an example of using the framework to generate the first shift Gray codes for weak orders in constant amortized time per string, where consecutive strings differ by a shift or a symbol change. [ABSTRACT FROM AUTHOR]

Let p be a prime number, in this paper, we investigate the structures of all constacyclic codes of length p s over the ring R u 2 , v 2 , p m = F p m u , v ∕ 〈 u 2 , v 2 , u v − v u 〉. The units of the ring R u 2 , v 2 , p m can be divided into following five forms: α , λ 1 = α + δ 1 u v , λ 2 = α + γ v + δ u v , λ 3 = α + β u + δ u v , λ 4 = α + β u + γ v + δ u v , where α , β , γ , δ 1 ∈ F p m ∗ and δ ∈ F p m . We obtain the algebraic structures of all constacyclic codes of length p s over R u 2 , v 2 , p m , except (α + δ 1 u v) -constacyclic codes, in terms of their polynomial generators and also find the number of codewords in each of these constacyclic codes. The number of constacyclic codes and duals of constacyclic codes corresponding to the units λ 2 , λ 3 and λ 4 are determined. We also provide examples to illustrate our results, which include several optimal codes. [ABSTRACT FROM AUTHOR]

Subspace codes have attracted much attention in recent years due to their applications to error correction in random network coding. In this paper, we construct several kinds of large cyclic subspace codes via Sidon spaces and large subspace codes via unions of some Sidon spaces. Therefore, some known results are extended. [ABSTRACT FROM AUTHOR]

Let R = GR (p e , m) [ u ] ∕ 〈 u k 〉 be a finite commutative ring for a prime p and any positive integers e , m and k. In this paper, we derive the explicit representation of cyclic codes over the ring R of length n , where n and p are coprime. We also discuss the dual of such cyclic codes over the ring R and give a sufficient condition for the codes to be self-dual. Moreover, we study quasi-cyclic codes of length k n and index k over the ring R , and obtain some good codes satisfying the bound given in Dougherty and Shiromoto (2000) over the ring Z 9 as an example. [ABSTRACT FROM AUTHOR]

The hulls of linear and cyclic codes over finite fields have been of interest and extensively studied due to their wide applications. In this paper, the hulls of cyclic codes of odd length n over the ring Z 4 have been focused on. Their characterization has been established in terms of the generators viewed as ideals in the quotient ring Z 4 [ x ] ∕ 〈 x n − 1 〉. An algorithm for computing the types of the hulls of cyclic codes of arbitrary odd length over Z 4 has been given. The 2-dimensions of the hulls of cyclic codes of length n over Z 4 and the number of cyclic codes of length n over Z 4 having hulls of a given 2-dimension are determined. The average 2-dimension E (n) of the hulls of cyclic codes of odd length n over Z 4 has been established. A general formula for E (n) has been established together with its upper and lower bounds. It turns out that E (n) grows the same rate as n. A brief discussion on hulls of cyclic codes over Z p 2 , where p is an odd prime, is provided as well. [ABSTRACT FROM AUTHOR]