Minimum Distance Of Code Research Articles

The minimum Hamming distances of repeated-root cyclic codes of length $$6p^s$$ and their MDS codes

In this paper, let p be a prime with $$p\ge 7$$ . We determine the weight distributions of cyclic codes of length 6 over $$\mathbb {F}_q$$ and the minimum Hamming distances of all repeated-root cyclic codes of length $$6p^s$$ over $$\mathbb {F}_q$$ , where q is a power of p and s is an integer. Furthermore, we find all maximum distance separable codes of length $$6p^s$$ .

Hamming and simplex codes for the sum-rank metric

Sum-rank Hamming codes are introduced in this work. They are essentially defined as the longest codes (thus of highest information rate) with minimum sum-rank distance at least 3 (thus one-error-correcting) for a fixed redundancy r, base-field size q and field-extension degree m (i.e., number of matrix rows). General upper bounds on their code length, number of shots or sublengths and average sublength are obtained based on such parameters. When the field-extension degree is 1, it is shown that sum-rank isometry classes of sum-rank Hamming codes are in bijective correspondence with maximal-size partial spreads. In that case, it is also shown that sum-rank Hamming codes are perfect codes for the sum-rank metric. Also in that case, estimates on the parameters (lengths and number of shots) of sum-rank Hamming codes are given, together with an efficient syndrome decoding algorithm. Duals of sum-rank Hamming codes, called sum-rank simplex codes, are then introduced. Bounds on the minimum sum-rank distance of sum-rank simplex codes are given based on known bounds on the size of partial spreads. As applications, sum-rank Hamming codes are proposed for error correction in multishot matrix-multiplicative channels and to construct locally repairable codes over small fields, including binary.

The Hamming distances of repeated-root cyclic codes of length [formula omitted

Due to the wide applications in consumer electronics, data storage systems and communication systems, cyclic codes have been an interesting research topic in coding theory. In this paper, let p be a prime with p≥7. We determine the Hamming distances of all repeated-root cyclic codes of length 5ps over Fq, where q=pm and both s and m are positive integers. Furthermore, we find all MDS cyclic codes of length 5ps and take quantum synchronizable codes from repeated-root cyclic codes of length 5ps. By comparing the minimum distances of 5ps-length repeated-root cyclic codes to BCH codes of close lengths, we illustrated that quantum synchronizable codes constructed from repeated-root cyclic codes have in general better performance in correcting Pauli errors than those from BCH codes.

Códigos de Reed-Muller

The Reed-Muller codes were discovered by David Eugene Muller and decoded by Irving Stoy Reed in 1954. Such codes belong to the linear code family and are widely used nowadays, mainly for their simple and efficient decoding algorithm. There are several ways to define Reed-Muller codes. In this work, we present, in a clear and simple way, a recursive definition for all Reed-Muller codes of order r \in N, denoted by R (r, m), where 0 <= r <= m, m \in N. Using this definition, we show the main parameters: length, number of elements and minimum distance of first-order Reed-Muller codes, R(1, m) for all m \in N. In addition, we present an application of the first-order codes in a National Aeronautics and Space Administration (NASA) space program.

Codes With Locality for Two Erasures

Codes with locality are a class of codes introduced by Gopalan et al. to efficiently repair a failed node, by minimizing the number of nodes contacted during repair. An $[n,k]$ systematic code is said to have information locality $r$ , if each message symbol can be recovered by accessing $\leq r$ other symbols. An $[n,k]$ code is said to have all-symbol locality $r$ , if each code symbol can be recovered by accessing $\leq r$ other symbols. In this paper, we consider a generalization of codes with all-symbol locality to the case of handling two erasures. We study codes with locality that can recover from two erasures via a sequence of two local, parity-check computations. We refer to these codes as sequential-recovery locally repairable codes (denoted by 2-seq LR codes). Earlier approaches to handling multiple erasures considered recovery in parallel; the sequential approach allows us to potentially construct codes with improved minimum distance. We derive an upper bound on the rate of 2-seq LR codes. We provide constructions based on regular graphs which are rate-optimal with respect to the derived bound. We also characterize the structure of any rate-optimal code. By studying the Generalized Hamming Weights of the dual code, we derive a recursive upper bound on the minimum distance of 2-seq LR codes. We also provide constructions of a family of codes based on Turan graphs, that are optimal with respect to this bound. We also present explicit distance-optimal Turan graph based constructions of 2-seq LR codes for certain parameters. Our approach also leads to a new bound on the minimum distance of codes with all-symbol locality for the single-erasure case.

New bounds on the minimum distance of cyclic codes

Two bounds on the minimum distance of cyclic codes are proposed. The first one generalizes the Roos bound by embedding the given cyclic code into a cyclic product code. The second bound also uses a second cyclic code, namely the non-zero-locator code, but is not directly related to cyclic product codes and it generalizes a special case of the Roos bound.

The implementation of L-codes in the system of residual classes

The possibilities of R-codes for error correction in the system SRC are being intensively investigated. This is due to the simplicity of the structure of R-codes and good corrective capabilities, as well as the comparative simplicity of their construction for any given minimum code distance. It is important and interesting to consider the so-called linear codes (L-codes) in the SRC. In the literature, L-codes are described qualitatively rather than quantitatively. Until now no one has researched in depth the properties of systems of residual classes, whose bases are not mutually prime numbers. Such a system also has certain corrective properties, which makes it necessary to assess the possibility and feasibility of using such systems to improve the reliability of computer systems and components. Therefore, this important and promising issue is considered in this article.

Some Applications of Coding Theory in the Projective Plane of Order Four

The main aim of this research is to introduce the relationship between the topic of coding theory and the projective plane of order four. The maximum value of size code M over the finite field of order four and an incidence matrix with the parameters, n (length of code), d (minimum distance of code) and e (error-correcting of code) have been constructed. Some examples and theorems have been given.

An Improved Bound on the Zero-Error List-Decoding Capacity of the 4/3 Channel

We prove a new upper bound on the size of codes $C \subseteq \{1,2,3,4\}^{n}$ with the property that every four distinct codewords in $C$ have a coordinate where they all differ. Specifically, we provide a self-contained proof that such codes have size at most $2^{6n/19 + o(n)}$ , that is, rate bounded asymptotically by 6/19 ≤ 0.3158 (measured in bits). This improves the previous best upper bound of 0.3512 due to (Arikan 1994), which in turn improved the 0.375 bound that followed from general bounds for perfect hashing due to (Fredman and Komlos, 1984) and (Korner and Marton, 1988). Finally, using a combination of our approach with a simple idea which exploits powerful bounds on the minimum distance of codes in the Hamming space, we further improve the upper bound to 0.31477.

New Bounds and Generalizations of Locally Recoverable Codes With Availability

We investigate the distance properties of linear locally recoverable codes (LRC codes) with all-symbol locality and availability. New upper and lower bounds on the minimum distance of such codes are derived. The upper bound is based on the shortening method and generalized Hamming weights that are fundamental parameters of any linear codes with many useful applications. This bound improves existing upper bounds. To reduce the gap in between upper and lower bounds, we do not restrict the alphabet size and propose explicit constructions of codes with locality and availability via rank-metric codes. The first construction relies on expander graphs and is better in low rate region. The second construction utilizes the LRC codes developed by Wang et al. as inner codes and is better in high rate region. We also suggest one possible generalization of LRC codes in which the recovering sets can intersect in a small number of coordinates. This feature allows us to increase the achievable code rate and still meet load balancing requirements. We derive upper and lower bounds on the parameters of such codes and present explicit constructions of codes with such a property.

On Distance Properties of Convolutional Polar Codes

A lower bound on the minimum distance of convolutional polar codes is provided. The bound is obtained from the minimum weight of the generalized cosets of the codes generated by the bottom rows of the polarizing matrix. Moreover, a construction of convolutional polar subcodes is proposed, which provides improved performance under successive cancellation list decoding. For sufficiently large list size, the decoding complexity of convolutional polar subcodes appears to be lower compared with Arikan polar subcodes with the same performance. The error probability of successive cancellation list decoding of convolutional polar subcodes is lower than that of Arikan polar subcodes with the same list size.

Multidimensional quasi-twisted codes: equivalent characterizations and their relation to multidimensional convolutional codes

We study multidimensional analogues of quasi-twisted codes from different points of view. Their concatenated structure allows us to characterize self-dual and complementary-dual classes of such codes as well as to show that multidimensional quasi-twisted (QT) codes are asymptotically good, together with their self-dual and complementary-dual subclasses. They are naturally related to nD convolutional codes as well. It is known that the minimum distance of quasi-cyclic codes provides a lower bound on the free distance of convolutional codes. An analogous result was shown for certain 1-generator 2D convolutional codes by using quasi-2D-cyclic codes. We prove a similar relation between convolutional codes and the related QT codes first, and then generalize the relation further to certain product convolutional codes and the related product QT codes, which improves the previous result in terms of dimension and number of generators. We also provide two-dimensional ternary and binary codes of modest lengths which yield good parameters.

Some q-ary Cyclic Codes from Explicit Monomials over $$\mathbb{F}_{q}m$$

Cyclic codes as a subclass of linear codes have practical applications in communication systems, consumer electronics, and data storage systems due to their efficient encoding and decoding algorithms. The objective of this paper is to construct some cyclic codes by the sequence approach. More precisely, we determine the dimension and the generator polynomials of three classes of q-ary cyclic codes defined by some sequences with explicit polynomials over \(\mathbb{F}_{q}m\). The minimum distance of such cyclic codes is also discussed. Some of these codes are optimal according to code tables. Moreover, the third class of cyclic codes provides some answers for Open Problem 3 proposed by Ding and Zhou in [1].

Algorithm 994

The minimum distance of an error-correcting code is an important concept in information theory. Hence, computing the minimum distance of a code with a minimum computational cost is crucial to many problems in this area. In this article, we present and assess a family of implementations of both the brute-force algorithm and the Brouwer-Zimmermann algorithm for computing the minimum distance of a random linear code over F 2 that are faster than current implementations, both in the commercial and public domain. In addition to the basic sequential implementations, we present parallel and vectorized implementations that produce high performances on modern architectures. The attained performance results show the benefits of the developed optimized algorithms, which obtain remarkable improvements compared with state-of-the-art implementations widely used nowadays.

On Two Classes of Primitive BCH Codes and Some Related Codes

BCH codes are an interesting type of cyclic codes and have wide applications in communication and storage systems. Generally, it is very hard to determine the minimum distances of BCH codes. In this paper, we determine the weight distributions of two classes of primitive BCH codes $\mathcal C_{(q, m, \delta _{2})}$ and $\mathcal C_{(q, m, \delta _{3})}$ and their extended codes, which solve two problems proposed by Ding et al. It is shown that the extended codes $\overline {\mathcal C}_{(q, m, \delta _{2})}$ have four nonzero weights. We also employ the Hartmann-Tzeng bound to present the minimum distance of the dual code $\mathcal C_{(q, m, \delta _{2})}^\perp $ for $q \ge 5$ . Inspired by the idea, we then determine the dimensions of a class of cyclic codes and give lower bounds on their minimum distances, which is greatly improved comparing with the BCH bound. Some optimal codes are obtained.

Binary additive MRD codes with minimum distance $$n-1$$ n - 1 must contain a semifield spread set

In this paper we prove a result on the structure of the elements of an additive maximum rank distance (MRD) code over the field of order two, namely that in some cases such codes must contain a semifield spread set. We use this result to classify additive MRD codes in $$M_n(\mathbb {F}_2)$$ with minimum distance $$n-1$$ for $$n\le 6$$ . Furthermore we present a computational classification of additive MRD codes in $$M_4(\mathbb {F}_3)$$ . The computational evidence indicates that MRD codes of minimum distance $$n-1$$ are much more rare than MRD codes of minimum distance n, i.e. semifield spread sets. In all considered cases, each equivalence class has a known algebraic construction.

From ds-Bounds for Cyclic Codes to True Minimum Distance for Abelian Codes

In this paper, we develop a technique to extend any bound for the minimum distance of cyclic codes constructed from its defining sets (ds-bounds) to Abelian (or multivariate) codes through the notion of ${\mathbb B}$ -apparent distance. We also study conditions for an Abelian code to verify that its ${\mathbb B}$ -apparent distance reaches its (true) minimum distance. Then, we construct some codes as an application.

A New Family of MRD Codes in <inline-formula> <tex-math notation="LaTeX">$\mathbb{F_q}^{2n\times2n}$ </tex-math> </inline-formula> With Right and Middle Nuclei <inline-formula> <tex-math notation="LaTeX">$\mathbb F_{q^n}$ </tex-math> </inline-formula>

In this paper, we present a new family of maximum rank distance (MRD for short) codes in $\mathbb F_{q}^{2n\times 2n}$ of minimum distance $2\leq d\leq 2n$. In particular, when $d=2n$, we can show that the corresponding semifield is exactly a Hughes-Kleinfeld semifield. The middle and right nuclei of these MRD codes are both equal to $\mathbb F_{q^n}$. We also prove that the MRD codes of minimum distance $2<d<2n$ in this family are inequivalent to all known ones. The equivalence between any two members of this new family is also determined.

Optimal Pliable Fractional Repetition Codes That are Locally Recoverable: A Bipartite Graph Approach

The main purpose of this paper is to construct pliable fractional repetition (FR) codes that are locally recoverable for distributed storage systems (DSSs). FR codes are integral in constructing a class of distributed storage codes with exact repair by transfer. Pliable FR codes are a new type of FR codes in which both the per-node storage and repetition degree can easily be adjusted simultaneously; thus, pliable FR codes are vital for DSSs in which parameters can dynamically change over time. However, the constructions of pliable FR codes with repair locality remain unknown. In addition, the tradeoffs between the code minimum distance of an FR code and its repair locality are not fully understood. To address these problems, this paper first presents general results regarding FR codes. Subsequently, this paper presents an improved Singleton-like bound for locally recoverable FR codes under an additional requirement that each node must be part of a local structure that, upon failure, allows it to be exactly recovered by a simple download process. Moreover, this paper proposes a construction of locally recoverable FR codes that can achieve the proposed Singleton-like bound; this construction is based on bipartite graphs with a given girth. In particular, this paper also proposes a general bipartite-graph-based approach to constructing optimal pliable FR codes with and without repair localities; in this approach, a new family of bipartite graphs, called matching-feasible graphs, is introduced. Finally, this paper proposes the explicit constructions of optimal pliable FR codes by using a family of matching-feasible graphs with arbitrary large girth. Notably, in addition to attaining a Singleton-like bound for FR codes, the explicit pliable FR codes are optimal locally recoverable FR codes from two perspectives of repair locality. The explicit pliable FR codes can also be used as FR batch codes to provide load balancing in DSSs.

Implementation of an algorithm to compute the strong apparent distance of bivariate codes

The BCH bound is the oldest lower bound for the minimum distance of a cyclic code. The study of this bound and its generalizations are classical topics, which includes the study of the very well-known family of BCH codes. In 1970, P. Camion extended the notion of BCH bound to the family of abelian codes by introducing the apparent distance of polynomials. Camion showed that the minimum value of the apparent distance of certain polynomials associated to codewords is less than or equal to the minimum distance of the code. The mentioned minimum value is known as the apparent distance of an abelian code. In 2016, Bernal-Bueno-Simón introduced the notion of strong apparent distance of polynomials and hypermatrices and developed an algorithm to compute the minimum strong apparent distance of a hypermatrix based on g-orbits manipulations. In this work, we will present the implementation of an algorithm to compute the strong apparent distance of bivariate codes.