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Motivated by applications in DNA storage, we study a setting in which strings are affected by tandem-duplication errors. In particular, we look at two settings: disjoint tandem-duplication errors, and equal-length tandem-duplication errors. We construct codes, with positive asymptotic rate, for the two settings, as well as for their combination. Our constructions are duplication-free codes, comprising codewords that do not contain tandem duplications of specific lengths. Additionally, our codes generalize previous constructions, containing them as special cases. [ABSTRACT FROM AUTHOR]

This paper presents a class of integer codes that are suitable for use in various optical networks. The presented codes are generated with the help of a computer and have the ability to correct l-bit burst errors corrupting one b-bit byte (1 ≤ l < b) and single errors corrupting two b-bit bytes. To evaluate the performance of the presented codes, we analyze their probabilities of incorrect decoding for two types of channels. In addition, the paper shows that the proposed codes can be interleaved without using an interleaver, which allows the decoder to correct all l-bit burst errors as well as many random errors. [ABSTRACT FROM AUTHOR]

We construct a class of F q F q [ u ] -additive constacyclic codes, where q = p m (p is an odd prime and m is a positive integer) and u 3 = u . We study their algebraic structure, and determine the asymptotic rates and relative distances of this class of codes. For the rates R of this class of codes, let 0 < δ < 1 3 and the q-ary entropy h q (2 δ) < 1 2 , there is a sequence of random F q F q [ u ] -additive constacyclic code C with R (C) converges to 1 4 with probability. Under such conditions, we further determine that the relative distance is convergent to δ with probability. We prove that F q F q [ u ] -additive constacyclic codes over finite non-chain ring F q [ u ] are asymptotically good [ABSTRACT FROM AUTHOR]

In this article, we find several novel and efficient quantum error-correcting codes ( Q ecc) by studying the structure of constacyclic ( C cc ), cyclic ( C c ), and negacyclic codes (N C c ) over the ring A k = Z p r 1 , r 2 , ⋯ , r k / ⟨ (r b (m b + 1) - r b) , r l r b = r b r l = 0 , b ≠ l ⟩ , where p = q m for m, m b ∈ N , m b | - 1 + q ∀ b , l ∈ 1 to k , q ≥ 3 is a prime, Z p is a finite field. We define distance-preserving gray map δ k . Moreover, we determine the quantum singleton defect (Q SD) of Q ecc, which indicates their overall quality. We compare our codes with existing codes in recent publications. The rings discussed by Kong et al. (EPJ Quantum Technol 10:1–16, 2023), Suprijanto et al. (Quantum codes constructed from cyclic codes over the ring F q + vF q + v 2 F q + v 3 F q + v 4 F q , pp 1–14, 2021. arXiv: 2112.13488v2 [cs.IT]), and Dinh et al. (IEEE Access 8:194082–194091, 2020) are specific cases of our work. Furthermore, we construct several novel and optimum linear complementary dual (Lcd) codes over A k. [ABSTRACT FROM AUTHOR]

Martínez-Peñas and Kschischang (IEEE Trans. Inf. Theory 65(8):4785–4803, 2019) proposed lifted linearized Reed–Solomon codes as suitable codes for error control in multishot network coding. We show how to construct and decode lifted interleaved linearized Reed–Solomon (LILRS) codes. Compared to the construction by Martínez-Peñas–Kschischang, interleaving allows to increase the decoding region significantly and decreases the overhead due to the lifting (i.e., increases the code rate), at the cost of an increased packet size. We propose two decoding schemes for LILRS that are both capable of correcting insertions and deletions beyond half the minimum distance of the code by either allowing a list or a small decoding failure probability. We propose a probabilistic unique Loidreau–Overbeck-like decoder for LILRS codes and an efficient interpolation-based decoding scheme that can be either used as a list decoder (with exponential worst-case list size) or as a probabilistic unique decoder. We derive upper bounds on the decoding failure probability of the probabilistic-unique decoders which show that the decoding failure probability is very small for most channel realizations up to the maximal decoding radius. The tightness of the bounds is verified by Monte Carlo simulations. [ABSTRACT FROM AUTHOR]

Digital signatures play a vital role in data security as they provide authenticity and non-repudiation of digital data. Code-based digital signatures are in high demand as quantum computers are extremely effective at breaking widely used digital signatures. The Courtois–Finiasz–Sendrier (CFS) scheme is one of the most popular code-based digital signature schemes. However, it has some disadvantages, such as a large public key size and poor signing efficiency. To address this issue, we construct a digital signature scheme named ENR DigiSig (Ekta Narwal and Rupali Digital Signature) using polar codes with several characteristics such as small signature size, low signing time, and high signing efficiency. Here, the hash of a shorter length is used in a specific way; then, padding is done to the hash output so that the result can be decoded. For this study, we have selected a fixed polar code rate of 0.5 and a blocklength of N = 2 n ; n ⩽ 4 . According to the experimental results, more than 96% of the signatures are generated successfully. [ABSTRACT FROM AUTHOR]

Let p be a prime number and q = p m for some positive integer m. In this paper, we find the possible Hermitian hull dimensions of λ -constacyclic codes over R e = F q 2 + u F q 2 + u 2 F q 2 + ⋯ + u e - 1 F q 2 , u e = 1 where F q 2 is the finite field of q 2 elements, e | (q + 1) and λ = η 1 α 1 + η 2 α 2 + ⋯ + η e α e for α l ∈ F q 2 ∗ of order r l such that r l ∣ q + 1 (for each 1 ≤ l ≤ e ). Further, we obtain some conditions for these codes to be Hermitian LCD. Also, under certain conditions, we establish a strong result that converts every constacyclic code to a Hermitian LCD code (Corollaries 2 and 3). We also study the structure of generator polynomials for Hermitian dual-containing constacyclic codes (Theorems 8 and 9), and obtain parameters of quantum codes using the Hermitian construction. The approach we used to derive Hermitian dual-containing conditions via the hull has not been used earlier. As an application, we obtain several optimal and near-to-optimal LCD codes, constacyclic codes having small hull dimensions, and quantum codes. [ABSTRACT FROM AUTHOR]

For an odd prime p and a positive integer r, let q = p r . The objective of this article is to study cyclic codes over the ring S = F q [ u , v , w ] / ⟨ u 3 - u , v 2 - v , w 2 - w , u v , v u , u w , w u , v w - w v ⟩ and to construct new and better quantum and LCD codes from them. We give the structure of cyclic codes over the ring S and obtain dual-containing codes over F q as the Gray images of dual-containing cyclic codes over S . Using these dual-containing codes, we obtain quantum codes and determine their parameters using the decomposition of cyclic codes over the ring S . We provide many new and better-than-existing quantum codes. We also give a method to obtain linear complementary dual (LCD) codes over S using the decomposition of cyclic codes over the ring S . We obtain some optimal and best-known linear codes as Gray images of LCD codes over S . [ABSTRACT FROM AUTHOR]

The sum-rank metric is a hybrid between the Hamming metric and the rank metric and suitable for error correction in multishot network coding and distributed storage as well as for the design of quantum-resistant cryptosystems. In this work, we consider the construction and decoding of folded linearized Reed–Solomon (FLRS) codes, which are shown to be maximum sum-rank distance (MSRD) for appropriate parameter choices. We derive an efficient interpolation-based decoding algorithm for FLRS codes that can be used as a list decoder or as a probabilistic unique decoder. The proposed decoding scheme can correct sum-rank errors beyond the unique decoding radius with a computational complexity that is quadratic in the length of the unfolded code. We show how the error-correction capability can be optimized for high-rate codes by an alternative choice of interpolation points. We derive a heuristic upper bound on the decoding failure probability of the probabilistic unique decoder and verify its tightness by Monte Carlo simulations. Further, we study the construction and decoding of folded skew Reed-Solomon codes in the skew metric. Up to our knowledge, FLRS codes are the first MSRD codes with different block sizes that come along with an efficient decoding algorithm. [ABSTRACT FROM AUTHOR]

Skew polynomials are a class of non-commutative polynomials that have several applications in computer science, coding theory and cryptography. In particular, skew polynomials can be used to construct and decode evaluation codes in several metrics, like e.g. the Hamming, rank, sum-rank and skew metric. We propose a fast divide-and-conquer variant of Kötter–Nielsen–Høholdt (KNH) interpolation algorithm: it inputs a list of linear functionals on skew polynomial vectors, and outputs a reduced Gröbner basis of their kernel intersection. We show, that the proposed KNH interpolation can be used to solve the interpolation step of interpolation-based decoding of interleaved Gabidulin codes in the rank-metric, linearized Reed–Solomon codes in the sum-rank metric and skew Reed–Solomon codes in the skew metric requiring at most O ~ s ω M (n) operations in F q m , where n is the length of the code, s the interleaving order, M (n) the complexity for multiplying two skew polynomials of degree at most n, ω the matrix multiplication exponent and O ~ · the soft-O notation which neglects log factors. This matches the previous best speeds for these tasks, which were obtained by top–down minimal approximant bases techniques, and complements the theory of efficient interpolation over free skew polynomial modules by the bottom-up KNH approach. In contrast to the top–down approach the bottom-up KNH algorithm has no requirements on the interpolation points and thus does not require any pre-processing. [ABSTRACT FROM AUTHOR]

As a tool towards quantum error correction, additive conjucyclic codes have gained great attention. But, their algebraic structure is completely unknown over finite fields (except F q 2 ) as well as rings. In this article, we investigate the structure of additive conjucyclic codes over Galois rings G R (2 r , 2) , where r ≥ 2 is an integer. We develop a one-to-one correspondence between the family of additive conjucyclic codes of length n over G R (2 r , 2) and the family of linear cyclic codes of length 2n over Z 2 r . This correspondence helps to obtain additive conjucyclic codes over G R (2 r , 2) via known linear cyclic codes over Z 2 r . We prove that the trace dual C Tr of an additive conjucyclic code C is also an additive conjucyclic code. Moreover, we derive a necessary and sufficient condition of additive conjucyclic codes to be self-dual. We further propose a technique for constructing linear cyclic codes over Z 2 r contained in additive conjucyclic codes over G R (2 r , 2) . Last but not least, we explicitly derive the generator matrices for these codes. [ABSTRACT FROM AUTHOR]

The error-correcting pair is a general algebraic decoding method for linear codes, which exists for many classical linear codes such as generalized Reed-Solomon codes. In this paper, we define a new extended generalized Reed-Solomon code, i.e., lengthened generalized Reed-Solomon code, which has good algebraic structure and many excellent properties, thus we extend the error-correcting pair to the case for lengthened generalized Reed-Solomon codes. Firstly, we give some sufficient conditions for which an LGRS code is non-GRS, and a necessary and sufficient condition for an LGRS code to be MDS or AMDS, respectively. And then, we constructively determine the existence of the error-correcting pair for lengthened generalized Reed-Solomon codes and give several examples to support our main results. [ABSTRACT FROM AUTHOR]

Recently, new studies on the decoding of cyclic codes have been developed. They place themselves under the umbrella of Cooper Philosophy and they consist in using (sparse) locator polynomials, which, once evaluated at the syndromes, return the error locations. In particular, it has been recently shown that it is not necessary to use Gröbner bases to compute such kind of polynomials, and that some sparse versions can be found (at least for error correction capability at most 2), using interpolation on the syndrome variety. In this paper, we study the combinatorial structure of the syndrome variety of a cyclic code and some of its variants, for error correction capability 2, by means of standard monomials. Such monomials can be found without computing a Gröbner basis of the syndrome ideal, neither performing any step of Buchberger reduction, that is, in a degröbnerized way. [ABSTRACT FROM AUTHOR]

A class of linear codes that extends classical Goppa codes to a non-commutative context is defined. An efficient decoding algorithm, based on the solution of a non-commutative key equation, is designed. We show how the parameters of these codes, when the alphabet is a finite field, may be adjusted to propose a McEliece-type cryptosystem. [ABSTRACT FROM AUTHOR]

We approach the study of complete bifix decodings of (uniformly) recurrent languages with the help of free profinite monoids. We show that the complete bifix decoding of a uniformly recurrent language F by an F-charged rational complete bifix code is uniformly recurrent. An analogous result is obtained for recurrent languages. As an application of the machinery developed within this approach, we show that the maximal pronilpotent quotient of the Schützenberger group of an irreducible symbolic dynamical system is an invariant of eventual conjugacy. [ABSTRACT FROM AUTHOR]

In this paper we review existing hard-decision decoding algorithms for product codes along with different post-processing techniques used in conjunction with the iterative decoder for product codes. We improve the decoder by Reddy and Robinson and use it to create a new post-processing technique. The performance of this new post-processing technique is evaluated through simulations, and these suggest that our new post-processing technique outperforms previously known post-processing techniques which are not tailored for specific codes. The cost of using the new post-processing technique is that the algorithm becomes more complex. However, the post-processing is applied very rarely unless the channel is very noisy, and hence the increase in computational complexity is negligible for most choices of parameters. Finally, we propose a new algorithm that combines existing techniques in a way that avoids the error floor with short relatively high rate codes. The algorithm should also avoid the error floor with long high rate codes, but further work is needed to confirm this. [ABSTRACT FROM AUTHOR]

For a prime p and a positive integer e, let F p e be the finite field and R : = F p e [ u , v ] / ⟨ f (u) , g (v) , u v - v u ⟩ , where f(u) and g(v) are non-constant square-free polynomials of degree r and s, respectively. This paper constructs quantum and linear complementary dual (briefly, LCD) codes from skew constacyclic codes over the ring R . Toward this, we first discuss the explicit structure of skew constacyclic codes and their Euclidean as well as Hermitian duals over R . Then, we establish a necessary and sufficient condition for these codes to contain their Euclidean (Hermitian) duals. Further, by applying CSS (Hermitian) construction, many new quantum codes with better parameters are obtained. Moreover, a necessary and sufficient condition is established for these codes over R to be Euclidean (Hermitian) LCD. Finally, many examples of MDS codes over F p e are provided under the gray images of the skew Euclidean LCD codes. [ABSTRACT FROM AUTHOR]

In this paper, we discuss double circulant and double negacirculant codes over a family of finite rings R t = F q + v 1 F q + v 2 F q + ⋯ + v t F q ; (v i 2 = v i , v i v j = v j v i = 0 , i , j = 1 , 2 , ... , t , i ≠ j) , where q is an odd prime power. We obtain necessary and sufficient conditions for double circulant codes (double negacirculant codes) to be self-dual codes and to be linear codes with complementary dual (or LCD codes) codes and study the algebraic structure of self-dual and LCD double circulant (double negacirculant codes) codes. We derive a formula to determine the total number of self-dual and LCD double circulant codes (double negacirculant codes) over the ring Rt. We also find distance bounds for double circulant codes over Rt. Moreover, we use a Gray map to prove that the families of self-dual and LCD double circulant codes under this Gray map are asymptotically good. [ABSTRACT FROM AUTHOR]

The maximum likelihood decoding problem (MLD) is known to be NP-hard and its complexity is strictly related to the security of some post-quantum cryptosystems, that is, the so-called code-based primitives. Analogously, the multivariate quadratic system problem (MQ) is NP-hard and its complexity is necessary for the security of the so-called multivariate-based primitives. In this paper we present a closed formula for a polynomial-time reduction from any instance of MLD to an instance of MQ, and viceversa. We also show a polynomial-time isomorphism between MQ and MLD, thus demonstrating the direct link between the two post-quantum cryptographic families. [ABSTRACT FROM AUTHOR]

In this paper, we present a class of integer codes capable of correcting l-bit burst errors within one b-bit byte and h-bit burst errors within two b-bit bytes (1 ≤ h < l < b). The presented codes are constructed with the help of a computer and have the potential to be used in various optical networks. To evaluate the performance of the proposed codes, we analyze the probability of incorrect decoding for different error rates. In addition, the paper shows that the presented codes are suitable for software implementation and that can be easily transformed into codes capable of correcting double burst errors. [ABSTRACT FROM AUTHOR]

In this paper, we study on linear, cyclic, self-dual and self-orthogonal codes over the ring Rq. We especially focus on extremal self-dual codes over F q . Our method is easy and constructing self-dual codes over F q . We define a distance-invariant Gray map from Rq to the finite field F q . It is proved that the images of self-dual codes of length n over Rq under this Gray map correspond to self-dual codes of length 2n over F q . Using all of these codes, we construct stabilizer quantum codes over F q . We obtain some self-dual codes and some optimal quantum codes. [ABSTRACT FROM AUTHOR]

In this paper, we study on linear, cyclic, self-dual and self-orthogonal codes over the ring Rq. We especially focus on extremal self-dual codes over F q . Our method is easy and constructing self-dual codes over F q . We define a distance-invariant Gray map from Rq to the finite field F q . It is proved that the images of self-dual codes of length n over Rq under this Gray map correspond to self-dual codes of length 2n over F q . Using all of these codes, we construct stabilizer quantum codes over F q . We obtain some self-dual codes and some optimal quantum codes. [ABSTRACT FROM AUTHOR]

In this paper, we present a class of integer codes capable of correcting burst asymmetric errors. The presented codes are constructed with the help of a computer and have the potential to be used in various practical systems, such as optical networks and VLSI memories. In order to evaluate the performance of the proposed codes, the paper analyzes the probability of erroneous decoding for different bit error rates. The presented codes are also analyzed from a rate-efficiency point of view. The obtained results show that for many data lengths they require less check-bits than optimal burst error correcting codes. [ABSTRACT FROM AUTHOR]

In this paper, we study constacyclic codes over the ring R k = F q [ u 1 , u 2 , ⋯ , u k ] / ⟨ u i 3 - u i , u i u j - u j u i ⟩ , for all i , j = 1 , ... , k , q = p m for any odd prime p and positive integers m, k. We study the structure of the ring R k and define a Gray map by a matrix and decompose constacyclic codes over the ring R k as a direct sum of constacyclic codes over F q . We give the relevant properties of constacyclic codes over the ring R k and give necessary and sufficient conditions for constacyclic codes to be dual-containing. Taking R 2 as an example, we give some corresponding conclusions. As an application, we construct some new quantum error-correcting codes over F q from constacyclic codes over R k . [ABSTRACT FROM AUTHOR]

This paper presents encoding and decoding algorithms for several families of optimal rank metric codes whose codes are in restricted forms of symmetric, alternating and Hermitian matrices. First, we show the evaluation encoding is the right choice for these codes and then we provide easily reversible encoding methods for each family. Later unique decoding algorithms for the codes are described. The decoding algorithms are interpolation-based and can uniquely correct errors for each code with rank up to ⌊(d − 1)/2⌋ in polynomial-time, where d is the minimum distance of the code. [ABSTRACT FROM AUTHOR]

Let F q be the finite field of order q = p m , where p is a prime, m is a positive integer, and R = F q [ u , v ] / ⟨ u 2 - u , v 2 - v , u v - v u ⟩ . Thus R [ x ; θ , δ θ ] is a noncommutative ring, known as skew polynomial ring, where θ is an automorphism of R and δ θ is a θ -derivation of R . The main concern of this work is to characterize (θ , δ θ) -cyclic codes over the ring R . Towards this, first we establish existence of the right division algorithm in R [ x ; θ , δ θ ] . Then we find generating polynomials and idempotent generators for (θ , δ θ) -cyclic codes over the ring R . Moreover, it is shown that (θ , δ θ) -cyclic codes are principally generated. Finally, by using the decomposition method, we have provided several examples of (θ , δ θ) -cyclic codes of different lengths over R out of them many are optimal as per the available database (Grassl, Code Tables: bounds on the parameters of various types of codes. http://www.codetables.de/). [ABSTRACT FROM AUTHOR]

In this paper, we present a class of integer code capable of correcting unidirectional solid bursts. A brief implementation in a quad core processor is discussed along with its memory consumption. Further, a comparison is done with linear codes and integer codes correcting similar types of errors. Codes discussed here use integers and table look ups which makes them suitable for software implementation in all processors. Also, these codes can be interleaved without the use of any additional hardware. [ABSTRACT FROM AUTHOR]

For a prime p and integer k > 1, we find the unique set of generators for cyclic codes over. Besides, the necessary and sufficient conditions for cyclic codes to be reversible are obtained. Finally, as an application, some computational examples are given in support of our results. [ABSTRACT FROM AUTHOR]

Let F q be a finite field of order q = p m , where p is an odd prime. This paper presents the study of self-dual and LCD double circulant codes over a class of finite commutative non-chain rings R q = F q + u F q + u 2 F q + ⋯ + u q - 1 F q where u q = u . Here, the whole contribution is two-folded. First, we enumerate self-dual and LCD double circulant codes of length 2n over R q , where n is an odd integer. Then by considering a dual-preserving Gray map ϕ , we show that Gray images of such codes are asymptotically good. Second, we investigate the algebraic structure of 1-generator quasi-cyclic (QC) codes over R q for q = 3 . In that context, we present their generator polynomials along with their minimal generating sets and minimum distance bounds. Here, it is proved that ϕ (C) is an sq-QC code of length nq over F q if C is an s-QC code of length n over R q . [ABSTRACT FROM AUTHOR]

For a prime p, let F q 2 be the finite field of q 2 elements where q = p m and m ≥ 1 is an integer. In this paper, we study constacyclic and skew constacyclic codes of length n over a class of finite commutative non-chain rings A q , r = F q 2 [ u 1 , u 2 , ⋯ , u r ] / ⟨ u i 2 - γ i u i , u i u j = u j u i = 0 ⟩ where 1 ≤ i ≠ j ≤ r , γ i ∈ F q 2 ∗ , and r ≥ 1 is an integer. For a unit Λ = ∑ i = 0 r κ i Λ i in the ring A q , r , we show that a (skew) Λ -constacyclic code of length n is a direct sum of (skew) Λ i -constacyclic codes of length n over F q 2 . Also, we derive the necessary and sufficient conditions for such codes (constacyclic and skew constacyclic) to contain their Hermitian duals. By applying the Hermitian construction on the Gray images of dual containing codes, many MDS and new quantum codes with parameters better than the best-known codes are constructed. [ABSTRACT FROM AUTHOR]

Coding theory and combinatorial t-designs have close connections and interesting interplay. One of the major approaches to the construction of combinatorial t-designs is the employment of error-correcting codes. As we all known, some t-designs have been constructed with this approach by using certain linear codes in recent years. However, only a small number of infinite families of cyclic codes holding an infinite family of 3-designs are reported in the literature. The objective of this paper is to study an infinite family of antiprimitive cyclic codes and determine their parameters. By the parameters of these codes and their dual, some infinite family of 3-designs are presented and their parameters are also explicitly determined. In particular, the complements of the supports of the minimum weight codewords in the studied cyclic code form a Steiner system. Furthermore, we show that the infinite family of cyclic codes admit 3-transitive automorphism groups. [ABSTRACT FROM AUTHOR]

Combinatorial t-designs are closely related to coding theory and finite geometries and have nice applications in some engineering areas. One of the major approaches to the construction of combinatorial t-designs is the employment of error-correcting codes. In this paper, this approach is further investigated with a class of linear codes related to the extended BCH codes and their duals over finite fields. By determining the parameters of these codes, an infinite family of 3-designs is presented and their parameters are also explicitly determined. In addition, the parameters of some combinatorial 2-designs from the extended codes of certain primitive BCH codes are also determined. Moreover, Research Problem 9.4 proposed by Ding in [Designs from Linear Codes, World Scientific, Singapore, pp: 257, 2018)] is also solved. [ABSTRACT FROM AUTHOR]

Wachter-Zeh (IEEE Trans Inf Theory 59(11):7268–7276, 2013), and later together with Raviv (IEEE Trans Inf Theory 62(4):1605–1615, 2016), proved that Gabidulin codes cannot be efficiently list decoded for any radius τ , providing that τ is large enough. Also, they proved that there are infinitely many choices of the parameters for which Gabidulin codes cannot be efficiently list decoded at all. Subsequently, in Trombetti and Zullo (IEEE Trans Inf Theory 66(9):5379–5386, 2020) these results have been extended to the family of generalized Gabidulin codes and to further family of MRD-codes. In this paper, we provide bounds on the list size of rank-metric codes containing generalized Gabidulin codes in order to determine whether or not a polynomial-time list decoding algorithm exists. We detect several families of rank-metric codes containing a generalized Gabidulin code as subcode which cannot be efficiently list decoded for any radius large enough and families of rank-metric codes which cannot be efficiently list decoded. These results suggest that rank-metric codes which are F q m -linear or that contains a (power of) generalized Gabidulin code cannot be efficiently list decoded for large values of the radius. [ABSTRACT FROM AUTHOR]

Error-correcting codes that can effectively encode and decode messages of distinct lengths while maintaining a constant blocklength are considered. It is known conventionally that a k-dimensional block code of length n defined over GF (q n) is designed to encode a k-symbol user data in to an n-length codeword, resulting in a fixed-rate coding. In contrast, considering q = p λ , this paper proposes two coding procedures (for the cases of λ = k and λ = n ) each deriving a multiple-rate code from existing channel codes defined over a composite field GF (q n) . Formally, the proposed coding schemes employ λ codes C 1 (λ , 1) , C 2 (λ , 2) , ... , C λ (λ , λ) defined over GF (q) to encode user messages of distinct lengths and incorporate variable-rate feature. Unlike traditional block codes, the derived multiple-rate codes of fixed blocklength n can be used to encode and decode user messages m of distinct lengths | m | = 1 , 2 , ... , k , k + 1 , ... , k n , thereby supporting a range of information rates—inclusive of the code rates 1 / n 2 , 2 / n 2 , ... , k / n 2 and 1 / n , 2 / n , ... , k / n ! A simple decoding procedure to the derived multiple-rate code is also given; in that, orthogonal projectors are employed for the identification of encoded user messages of variable length. [ABSTRACT FROM AUTHOR]

Let C be a code with parity-check matrix H. A stopping set S of size l ≤ n for H is an l-columns submatrix of Hs of H which dose not contain a row with weight one. In this paper we consider the code which parity-check matrix is incidence matrix of complete graph Km. These codes are LDPC codes and we obtain the stopping sets for these codes. [ABSTRACT FROM AUTHOR]

In this article, for a prime p such that p ≡ 2 or 3 (mod 5) , we identify cyclic codes of length N over R = M 2 (F p + u F p) , u 2 = 0 as right R-submodules of R [ x ] / ⟨ x N - 1 ⟩ . Also, we define an isometry from M 2 (F p + u F p) to F p 2 + u F p 2 + v F p 2 + u v F p 2 , where u 2 = v 2 = 0 , u v = v u and determine the structure of cyclic codes, in particular self-dual cyclic codes of length N where gcd (N , p) = 1 . Moreover, several optimal and near to optimal codes are obtained as the Gray images of these codes over R. [ABSTRACT FROM AUTHOR]

For an odd prime p and q = pr, this paper deals with LCD codes obtained from cyclic codes of length n over a finite commutative non-chain ring R = F q [ u , v ] / 〈 u 2 − α u , v 2 − 1 , u v − v u 〉 where α is a non-zero element in F q . Initially, we impose certain conditions on the generator polynomials of cyclic codes when gcd (n , p) = 1 and gcd (n , p) ≠ 1 , respectively so that these codes become LCD. Then, by defining a Gray map ψ, we show that the Gray image of an LCD code of length n over R is an LCD code of length 4n over F q . In this way, we obtain many optimal and best-known linear codes (BKLC) from the Gray images of both cyclic and LCD codes over R . Eventually, by applying the CSS construction on cyclic codes over R that contain their Euclidean duals, we determine many superior quantum codes compared to the existing codes in the recent references. [ABSTRACT FROM AUTHOR]

Let q be an odd prime power, and denote by F q the finite field with q elements. In this paper, we consider the ring R = F q + u F q + v F q , where u 2 = u , v 2 = v , u v = v u = 0 and study double circulant and double negacirculant codes over this ring. We first obtain the necessary and sufficient conditions for a double circulant code to be self-dual (resp. LCD). Then we enumerate self-dual and LCD double circulant and double negacirculant codes over R. Last but not the least, we show that the family of Gray images of self-dual and LCD double circulant codes over R are good. Several numerical examples of self-dual and LCD codes over F 5 as the Gray images of these codes over R are given in short lengths. [ABSTRACT FROM AUTHOR]

Following the work of Gaborit et al. (in: The international workshop on coding and cryptography (WCC 13), 2013) defining LRPC codes over finite fields, Renner et al. (in: IEEE international symposium on information theory, ISIT 2020, 2020) defined LRPC codes over the ring of integers modulo a prime power, inspired by the paper of Kamche and Mouaha (IEEE Trans Inf Theory 65(12):7718–7735, 2019) which explored rank metric codes over finite principal ideal rings. In this work, we successfully extend the work of Renner et al. by constructing LRPC codes over the ring Z m which is not a chain ring. We give a decoding algorithm and we study the failure probability of the decoder. [ABSTRACT FROM AUTHOR]