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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 .

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.

Let R be a Galois ring of characteristic p a , where p is a prime and a is a natural number. In this paper, the generators of cyclic codes of arbitrary length n over R in terms of minimal degree polynomials of certain subsets of codes have been obtained. Moreover, the explicit set of generators so obtained turns out to be a minimal strong Grobner basis. Some results on torsion codes of a cyclic code over R have also been obtained. Using these results, the size of a cyclic code over R has been expressed in terms of the degrees of its generators.

Steganographic schemes are used in many fields, as it provides high-security transmission of messages. The strength of the steganographic scheme depends on embedding efficiency. Embedding efficiency is understood as the average number of random data bits that must be embedded in an object per one embedding change. The below-done work introduces an efficient embedding technique using various linear block codes useful to both spatial and frequency domain of digital images. Linear block coding algorithms like binary hamming code, Random linear code, Cyclic code, Reed Solomon code, etc., are applied to embed restricted data inside the image, which improves embedding efficiency. Higher embedding efficiency translates to better steganographic security. The Mean squared error (MSE), Peak Signal to noise ratio (PSNR), Structural Similarity Index (SSIM), Normalized Absolute Error (NAE) and Correlation Coefficient values for each algorithm is calculated and compared. Histogram attack is one of the steganalysis techniques which is used in this paper to confirm the security provided to the restricted data. This comparison shows that Binary Hamming code in the spatial domain and transform domain provides better results as compared to the other coding theory algorithms.

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 .

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 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.

Linear codes with complementary-duals (LCD) have many applications in cryptography, communication systems and data storage. A q-ary linear code C is called LCD if C∩C⊥ = {0} holds. Using method of coset theory, we deduce a characterization of LCD cyclic code by its defining set. Then two families of q-ary MDS cyclic LCD codes with lengthes n| (q+1) and n| (q-1) are determined and many new classes of LCD MDS codes are gained.

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 .

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 .

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.

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 .

Spectral Amplitude Coding – Optical Code Division Multiple Access systems are gaining popularity due to their property of very low multiple access interference. A new code with zero cross correlation, termed as Zero Cross Correlation Code (ZCCC) is proposed. Code construction algorithm is designed with any weight for any number of users having constant, or variable weights. Variable weight (VW) codes give different quality of service, and are suitable, and useful for multimedia applications. The performance of proposed codes is analyzed using direct detection technique. Numerical results show that the use of proposed code yields lower bit error rate, and/or require lower code length than existing codes. Simulation results on OptiSystem 13 also show better performance of proposed code than existing ZCC code (Anuar et al., 2009). Proposed ZCCC requires only two filtering elements irrespective of weight, and construct codes without using mapping as compared to ZCC codes.

A characterization of a class of optimal three-weight cyclic codes of dimension 3 over any finite field was recently presented in [10]. Shortly after this, a generalization for the sufficient numerical conditions of such characterization was given in [3]. The main purpose of this work is to show that the numerical conditions found in [3], are also necessary. As we will see later, an interesting feature of the present work, in clear contrast with these two preceding works, is that we use some new and non-conventional methods in order to achieve our goals. In fact, through these non-conventional methods, we not only were able to extend the characterization in [10], but also present a less complex proof of such extended characterization, which avoids the use of some of the sophisticated --but at the same time complex-- theorems, that are the key arguments of the proofs given in [10] and [3]. Furthermore, we also find the parameters for the dual code of any cyclic code in our extended characterization class. In fact, after the analysis of some examples, it seems that such dual codes always have the same parameters as the best known linear codes., Comment: arXiv admin note: text overlap with arXiv:1508.05077

A new two dimensional Diagonal Eigenvalue Unity (2D-DEU) code is developed for the spectral⧹spatial optical code division multiple access (OCDMA) system. It has a lower cross correlation value compared to two dimensional diluted perfect difference (2D-DPD), two dimensional Extended Enhanced Double Weight (2D-Extended-EDW) codes. Also, for the same code length, the number of users can be generated by the 2D-DEU code is higher than that provided by the others codes. The Bit Error Rate ( BER ) numerical analysis is developed by considering the effects of shot noise, phase induced intensity noise (PIIN), and thermal noise. The main result shows that BER is strongly affected by PIIN for the higher source power. The 2D-DEU code performance is compared with 2D-DPD, 2D-Extended-EDW and two dimensional multi-diagonals (2D-MD) codes. This comparison proves that the proposed 2D-DEU system outperforms the related codes.

The interplay between coding theory and t -designs started many years ago. While every t -design yields a linear code over every finite field, the largest t for which an infinite family of t -designs is derived directly from a linear or nonlinear code is t = 3 . Sporadic 4 -designs and 5 -designs were derived from some linear codes of certain parameters. The major objective of this paper is to construct many infinite families of 2 -designs and 3 -designs from linear codes. The parameters of some known t -designs are also derived. In addition, many conjectured infinite families of 2 -designs are also presented.

BCH codes have been studied for over fifty years and widely employed in consumer devices, communication systems, and data storage systems. However, the dimension of BCH codes is settled only for a very small number of cases. In this paper, we study the dimensions of BCH codes over finite fields with three types of lengths n , namely n = q m − 1 , n = ( q m − 1 ) ∕ ( q − 1 ) and n = q m + 1 . For narrow-sense primitive BCH codes with designed distance δ , we investigate their dimensions for δ in the range 1 ≤ δ ≤ q ⌈ m 2 ⌉ + 1 . For non-narrow sense primitive BCH codes, we provide two general formulas on their dimensions and give the dimensions explicitly in some cases. Furthermore, we settle the minimum distances of some primitive BCH codes. We also explore the dimensions of the BCH codes of lengths n = ( q m − 1 ) ∕ ( q − 1 ) and n = q m + 1 over finite fields.

Bierbrauer (2012) developed the theory of q-linear cyclic codes over (Fq)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,m), where m is a prime integer. Let R be a finite commutative ring and Rn=R[x]xn1. In this paper, we generalize the theory of Fq-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 Rn-linear code. Using the structure of Rn-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 (Fq)m are described by ring-theoretic facts, and the structure of cyclic additive codes over the Galois ring GR(p,m) is given for an arbitrary integer m, not necessarily a prime number.

In this paper, a new two-dimensional spectral/spatial codes family, named two dimensional dynamic cyclic shift codes (2D-DCS) is introduced. The 2D-DCS codes are derived from the dynamic cyclic shift code for the spectral and spatial coding. The proposed system can fully eliminate the multiple access interference (MAI) by using the MAI cancellation property. The effect of shot noise, phase-induced intensity noise and thermal noise are used to analyze the code performance. In comparison with existing two dimensional (2D) codes, such as 2D perfect difference (2D-PD), 2D Extended Enhanced Double Weight (2D-Extended-EDW) and 2D hybrid (2D-FCC/MDW) codes, the numerical results show that our proposed codes have the best performance. By keeping the same code length and increasing the spatial code, the performance of our 2D-DCS system is enhanced: it provides higher data rates while using lower transmitted power and a smaller spectral width.

Let α be a primitive element of a finite field Fr, where r=qm1m2 and gcd⁡(m1,m2)=d, so α1=αr−1qm1−1 and α2=αr−1qm2−1 are primitive elements of Fqm1 and Fqm2, respectively. Let e be a positive integer such that gcd⁡(e,qm2−1qd−1)=1, Fqm2=Fq(α2e), and α1 and α2e are not conjugates over Fq. We define a cyclic code C={c(a,b):a∈Fqm1,b∈Fqm2}, c(a,b)=(T1(aα1i)+T2(bα2ei))i=0n−1, where Ti denotes the trace function from Fqmi to Fq for i=1,2. In this paper, we use Gauss sums to investigate the weight distribution of C, which generalizes the results of C. Li and Q. Yue in [13,14]. Furthermore, we explicitly determine the weight distribution of C if d=1,2. Moreover, we prove it is optimal three-weight achieving the Griesmer bound if d=1 and gcd⁡(m2−em1,q−1)=1.

We completely determine the minimum Lee weights of cyclic self-dual codes over a Galois ring G R ( p 2 , m ) of length p k , where m and k are positive integers and p is a prime number. We obtain all cyclic self-dual codes over G R ( 2 2 , 1 ) ≅ Z 4 of lengths 16 and 32 with their Lee weight enumerators. We also find cyclic self-dual codes over G R ( 3 2 , 1 ) ≅ Z 9 (respectively, G R ( 3 2 , 2 ) ) of lengths up to 27 (respectively, 9). Most of the cyclic self-dual codes we found are extremal with respect to the Lee weights.

Necessary and Sufficient conditions for an equidistant code to be a 2-TA code are obtained. An explicit construction method is proposed to obtain linear MDS p + 1 , 2 , p codes over the finite field F p , where p is a prime. These codes can be used as 2-TA codes for p 2 . In particular, for p = 3 , it is observed that the linear 4, 2, 3 MDS code contradicts a result of Jin and Blaum (2007). The correct version of this result and its proof is given. Existence of some infinite families of equidistant 2-TA codes is shown by using Jacobsthal and Hadamard matrices. Some of these codes are also observed to be good equidistant code (Sinha et al., 2008).

In this paper we present a construction for S-boxes using quasi-cyclic codes. We obtain S-boxes with good nonlinearity.

In this paper a generalization of classic Mollard construction for any code length is given. It is shown that such generalized codes have property of partial robustness. This construction has less undetectable and miscorrected errors than the traditional linear error-correcting codes, just as than the generalized Vasil'ev code for some code parameters, thereby providing better protection against multiple bit errors.

The structure of λ-constacyclic codes of length 2 s over the Galois ring GR ( 2 a , m ) is obtained, for any unit λ of the form 4 z − 1 , z ∈ GR ( 2 a , m ) . The duals codes and necessary and sufficient conditions for the existence of a self-dual λ-constacyclic code are provided. Among others, this structure is used to establish the Hamming, homogeneous, and Rosenbloom–Tsfasman distances, and Rosenbloom–Tsfasman weight distribution of all such constacyclic codes.

In this paper, a class of binary cyclic codes with three generalized Niho-type nonzeros is introduced. Based on some techniques in solving certain equations over finite fields, the proposed cyclic codes are shown to have six nonzero weights and the weight distribution is also completely determined.

Let F 2 m be the finite field of 2 m elements and s be any positive integer. The existing literature only gives an effective calculation method to represent all distinct Euclidean self-dual cyclic codes of length 2 s over the finite chain ring F 2 m + u F 2 m ( u 2 = 0 ) , such as in Cao et al., (2019). As a development of this topic, we provide an explicit expression for each of these self-dual cyclic codes, using binomial coefficients. The Gray image of any self-dual cyclic code over F 2 m + u F 2 m of length 2 s is a self-dual 2 -quasi-cyclic code over F 2 m of length 2 s + 1 . In particular, we give a generator matrix for each of these self-dual 2 -quasi-cyclic codes over F 2 m .

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.

Let R be a finite commutative chain ring with unique maximal ideal { γ } , and let n be a positive integer relatively prime to the characteristic of R / { γ } . In this paper, some new necessary and sufficient conditions for the existence of nontrivial self-dual cyclic codes of length n over R are provided. Explicit enumeration formulas for self-dual cyclic codes over R are studied. In particular, several main results of 21 are special cases of the present paper.

Two-dimensional cyclic code is one of the natural generalizations of cyclic code. In this paper we study the algebraic structure of some two-dimensional cyclic codes and their dual codes.

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.

A new chain code named Unsigned Manhattan Chain Code (UMCC) is presented.UMCC is insensitive to rotation and mirroring, and enables shape magnification.UMCC is capable of detecting monotone parts of the shape's boundary.The proposed chain code achieves 50% better compression ratio than F8 code. This paper introduces a new chain code named Unsigned Manhattan Chain Code - UMCC. Although it exploits a pixel's neighbourhood of 8-connectivity, only two coding symbols are used for navigating through the geometric shape's boundary pixels. For this, the movements in the x - and y -coordinate direction are separated, whilst the sign of the moving direction is controlled by two flags - one for each coordinate direction. UMCC is insensitive to rotation and mirroring and enables shape magnification. However, the more unique property is the UMCC's ability to explicitly separate the monotonic parts of the geometric shape. UMCC's properties have been compared against the properties of other chain codes including the Freeman chain code in eight and four directions, the Vertex Chain Code, and the Three OrThogonal chain code. It has been shown that the UMCC has superior properties in regards to the up-to-date chain codes.

Double Weight (DW) code family is one of the coding schemes proposed for Spectral Amplitude Coding–Optical Code Division Multiple Access (SAC–OCDMA) systems. Modified Double Weight (MDW) code for even weights and Enhanced Double Weight (EDW) code for odd weights are two algorithms extending the use of DW code for SAC–OCDMA systems. The above mentioned codes use mapping technique to provide codes for higher number of users. A new generalized algorithm to construct EDW and MDW like codes without mapping for any weight greater than 2 is proposed. A single code construction algorithm gives same length increment, Bit Error Rate ( BER ) calculation and other properties for all weights greater than 2. Algorithm first constructs a generalized basic matrix which is repeated in a different way to produce the codes for all users (different from mapping). The generalized code is analysed for BER using balanced detection and direct detection techniques.

Let R = Z 4 be the integer ring mod 4. A double cyclic code of length ( r , s ) over R is a set that can be partitioned into two parts that any cyclic shift of the coordinates of both parts leaves invariant the code. These codes can be viewed as R x -submodules of R x / ( x r - 1 ) ? R x / ( x s - 1 ) . In this paper, we determine the generator polynomials of this family of codes as R x -submodules of R x / ( x r - 1 ) ? R x / ( x s - 1 ) . Further, we also give the minimal generating sets of this family of codes as R-submodules of R x / ( x r - 1 ) ? R x / ( x s - 1 ) . Some optimal or suboptimal nonlinear binary codes are obtained from this family of codes. Finally, we determine the relationship of generators between the double cyclic code and its dual.

A matrix-product structure for repeated-root cyclic codes over finite fields is explored. Using this, some properties such as minimum distance and duality for these codes are rediscovered. Finally, a decoding algorithm is presented for this class of codes and the algorithm is modified and adapted for the binary repeated-root cyclic codes of length 2 k .

In this paper, we study cyclic codes and constacyclic codes with shift constant ( 2 + u ) over R = Z 4 + u Z 4 , where u 2 = 1 . We determine the form of the generators of the cyclic codes over this ring and their spanning sets. Considering their Z 4 images, we prove that the Z 4 -image of a ( 2 + u ) -constacyclic code of odd length is a cyclic code over Z 4 . We also present many examples of cyclic codes over R whose Z 4 -images have better parameters than previously best-known Z 4 -linear codes.

Let D 2 n be the dihedral group of order n. Left ideals of the group algebra F q D 2 n are known as left dihedral codes over F q of length 2n, and abbreviated as left D 2 n -codes. In this paper, a system theory for left D 2 n -codes is developed only using finite field theory and basic theory of cyclic codes and skew cyclic codes. First, we prove that any left D 2 n -code is a direct sum of concatenated codes with inner codes A i and outer codes C i , where A i is a minimal self-reciprocal cyclic code over F q of length n and C i is a skew cyclic code of length 2 over an extension field or principal ideal ring of F q . Then for the case of gcd ? ( n , q ) = 1 , we give a precise description for outer codes in the concatenated codes, provide the dual code for any left D 2 n -code and determine all self-dual left D 2 n -codes. Moreover, all 1995 binary left dihedral codes and all 255 binary self-dual left dihedral codes of length 30 are given, and a class of left D 2 p n -codes over F q is investigated. We develop a system theory for left dihedral codes only using finite field theory, basic theory of cyclic codes and skew cyclic codes.We prove that any left dihedral code is a direct sum of concatenated codes in which the inner codes and outer codes are cyclic codes and skew cyclic codes respectively.We provide a precise expression for all distinct left D 2 n -code over F q , where D 2 n is the dihedral group of order n and gcd ? ( n , q ) = 1 .We give the dual code of any left D 2 n -code and list all self-dual left D 2 n -codes explicitly.

In this paper we find the complete weight hierarchy of the codes parameterized by the edges of the cycle C 4 and an upper bound in the case of the complete bipartite graph K 2 , n . Moreover we find some bounds for the generalized Hamming weights of some codes parameterized by a set of monomials of the same degree. These bounds work in the case of codes parameterized by the edges of any simple graph G . Also we establish the relationships among the generalized Hamming weights of any parameterized code of the form C X ( d ) and the generalized Hamming weights of the parameterized code of the form C X ' ( d ) , when X ' is embedded in X .

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. Let $m=2\ell+1$ for an integer $\ell\geq 1$ and $\pi$ be a generator of $\gf(3^m)^*$. In this paper, a class of cyclic codes $\C_{(u,v)}$ over $\gf(3)$ with two nonzeros $\pi^{u}$ and $\pi^{v}$ is studied, where $u=(3^m+1)/2$, and $v=2\cdot 3^{\ell}+1$ is the ternary Welch-type exponent. Based on a result on the non-existence of solutions to certain equation over $\gf(3^m)$, the cyclic code $\C_{(u,v)}$ is shown to have minimal distance four, which is the best minimal distance for any linear code over $\gf(3)$ with length $3^m-1$ and dimension $3^m-1-2m$ according to the Sphere Packing bound. The duals of this class of cyclic codes are also studied.

A recovering set for a coordinate position i in a code is a set R i of other coordinate positions such that the value at the i th position can be recovered by accessing the values at coordinate positions in R i . A locally recoverable (LRC) code with multiple recovering sets is a code in which for every coordinate position there are more than one recovering set. Such codes have been generally studied with the assumption that the recovering sets are pairwise disjoint. Recently Kruglik et al. [1] have presented a construction of LRC codes in which the recovering sets of a coordinate need not be disjoint. In this paper, we present a construction of such type of codes by using the construction of RS-like LRC codes. Further, using a bound given in [1] , we have obtained a bound on the rate of the codes from the present construction. Also, we have presented a sufficient condition for a cyclic code over a finite field to be an LRC code with intersecting recovering sets.

Let p be any odd prime number, m and s be arbitrary positive integers, and let F p m be the finite field of cardinality p m . Existing literature only determines the number of all (Euclidean) self-dual cyclic codes of length p s over finite chain ring R = F p m + u F p m ( u 2 = 0 ) , such as Dinh et al. (2018). Using some combinatorial identities, we obtain certain properties for Kronecker product of matrices over F p with a specific type. On that basis, we give an explicit representation and enumeration for all distinct self-dual cyclic codes of length p s over R . Moreover, we provide an efficient method to construct every self-dual cyclic code of length p s over R precisely.

Cyclic code is an interesting topic in coding theory and communication systems. In this paper, we investigate the ternary cyclic codes with parameters [ 3 m − 1 , 3 m − 1 − 2 m , 4 ] based on some results proposed by Ding and Helleseth in 2013. Six new classes of optimal ternary cyclic codes are presented by determining the solutions of certain equations over F 3 m .

We study α -constacyclic codes over the Frobenius non-chain ring R ≔ Z 4 [ u ] ∕ 〈 u 2 − 1 〉 for any unit α of R . We obtain new MDR cyclic codes over Z 4 using a close connection between α -constacyclic codes over R and cyclic codes over Z 4 . We first explicitly determine generators of all α -constacyclic codes over R of odd length n for any unit α of R . We then explicitly obtain generators of cyclic codes over Z 4 of length 2 n by using a Gray map associated with the unit α . This leads to a construction of infinite families of MDR cyclic codes over Z 4 , where a MDR code means a maximum distance with respect to rank code in terms of the Hamming weight or the Lee weight. We obtain 202 new cyclic codes over Z 4 of lengths 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50 and 54 by implementing our results in Magma software; some of them are also MDR codes with respect to the Hamming weight or the Lee weight.

A novel design of all optical binary to gray code converter is proposed and described. Gray code is a cyclic code, in which predecessor and successor codes have one bit change. So these codes are suitable for error detection in the optical communication. This device consists of Mach-Zehnder interferometer as switching element, having the ability to switch the light from one port to other port at ultra-fast rate. The performance parameters like extension ratio and insertion ratio are computed and obtained as 20.59 dB and 0.951 dB respectively. Further the study is verified using finite difference time division method. The results are also verified mathematically and with MATLAB.

Let F 2 m be a finite field of cardinality 2 m , R = F 2 m + u F 2 m ( u 2 = 0 ) and s , n be positive integers such that n is odd. In this paper, we give an explicit representation for every self-dual cyclic code over the finite chain ring R of length 2 s n and provide a calculation method to obtain all distinct codes. Moreover, we obtain a clear formula to count the number of all these self-dual cyclic codes. As an application, self-dual and 2-quasi-cyclic codes over F 2 m of length 2 s + 1 n can be obtained from self-dual cyclic code over R of length 2 s n and by a Gray map preserving orthogonality and distances from R onto F 2 m 2 .

A variable–weight code family referred to as variable–weight quadratic congruence code (VWQCC) is constructed by algebraic transformation for incoherent synchronous optical code division multiple access (OCDMA) systems. Compared with quadratic congruence code (QCC), VWQCC doubles the code cardinality and provides the multiple code-sets with variable code–weight. Moreover, the bit-error rate (BER) performance of VWQCC is superior to those of conventional variable–weight codes by removing or padding pulses under the same chip power assumption. The experiment results show that VWQCC can be well applied to the OCDMA with quality of service (QoS) requirements.

Let Fq be the finite field with q elements and Cn be the cyclic group of order n, where n is a positive integer relatively prime to q . Let H,K be subgroups of Cn such that H is a proper subgroup of K. In this note, the weight distributions of the cyclic codes of length n over Fq with generating idempotents View the MathML source and View the MathML source are explicitly determined, where View the MathML source and View the MathML source. Our result naturally gives a new characterization of a theorem by Sharma and Bakshi [18] that determines the weight distribution of all irreducible cyclic codes of length pm over Fq, where p is an odd prime and q is a primitive root modulo pm. Finally, two examples are presented to illustrate our results. Accepted version

Let F q be a finite field with q elements and m 1 , m 2 two distinct positive integers such that gcd ? ( m 1 , m 2 ) = d . Suppose that α 1 and α 2 are two primitive elements of F q m 1 and F q m 2 , respectively. Let n = ( q m 1 - 1 ) ( q m 2 - 1 ) / ( q d - 1 ) and T i denote the trace function from F q m i to F q for i = 1 , 2 . We define a cyclic code C ( q , m 1 , m 2 ) = { c ( a , b ) : a ? F q m 1 , b ? F q m 2 } , where c ( a , b ) = ( T 1 ( a α 1 0 ) + T 2 ( b α 2 0 ) , T 1 ( a α 1 1 ) + T 2 ( b α 2 1 ) , ? , T 1 ( a α 1 n - 1 ) + T 2 ( b α 2 n - 1 ) ) . In this paper, we use Gauss sums to investigate the weight distribution of C ( q , m 1 , m 2 ) and prove that it has at most four nonzero weights if d = 2 and gcd ? ( m 1 - m 2 2 , q - 1 ) = 1 . Furthermore, we get a class of three-weight cyclic codes if | m 1 - m 2 | = 2 . Some optimal or nearly optimal cyclic codes are presented.