Minimum Distance Of Linear Codes Research Articles

Fast parallel computation of reduced row echelon form to find the minimum distance of linear codes

Fast parallel computation of reduced row echelon form to find the minimum distance of linear codes

On construction of quantum codes with dual-containing quasi-cyclic codes

One of the main objectives of quantum error-correction theory is to construct quantum codes with optimal parameters and properties. In this paper, we propose a class of 2-generator quasi-cyclic codes and study their applications in the construction of quantum codes over small fields. Firstly, some sufficient conditions for these 2-generator quasi-cyclic codes to be dual-containing concerning Hermitian inner product are determined. Then, we utilize these Hermitian dual-containing quasi-cyclic codes to produce quantum codes via the famous Hermitian construction. Moreover, we present a lower bound on the minimum distance of these quasi-cyclic codes, which is helpful to construct quantum codes with larger lengths and dimensions. As the computational results, many new quantum codes that exceed the quantum Gilbert-Varshamov bound are constructed over $F_q$, where $q$ is $2,3,4,5$. In particular, 16 binary quantum codes raise the lower bound on the minimum distance in Grassl's table \cite{Grassl:codetables}. In nonbinary cases, many quantum codes are new or have better parameters than those in the literature.

Mutation-Based Algebraic Artificial Bee Colony Algorithm for Computing the Distance of Linear Codes

Finding the minimum distance of linear codes is a non-deterministic polynomial-time-hard problem and different approaches are used in the literature to solve this problem.
 Although, some of the methods focus on finding the true distances by using exact algorithms, some of them focus on optimization algorithms to find the lower or upper bounds of the distance. In this study,
 we focus on the latter approach. We first give the swarm intelligence background of artificial bee colony algorithm, we explain the algebraic approach of such algorithm and call it the algebraic artificial bee colony algorithm (A-ABC). Moreover, we develop the A-ABC algorithm by integrating it with the algebraic differential mutation operator. We call the developed algorithm the mutation-based algebraic artificial bee colony algorithm (MBA-ABC). We apply both; the A-ABC and MBA-ABC algorithms to the problem of finding the minimum distance of linear codes. The achieved results indicate that the MBA-ABC algorithm has a superior performance when compared with the A-ABC algorithm when finding the minimum distance of Bose, Chaudhuri, and Hocquenghem (BCH) codes (a special type of linear codes).

On the generalization of the construction of quantum codes from Hermitian self-orthogonal codes

Many q-ary stabilizer quantum codes can be constructed from Hermitian self-orthogonal q^2-ary linear codes. This result can be generalized to q^{2 m}-ary linear codes, m > 1. We give a result for easily obtaining quantum codes from that generalization. As a consequence we provide several new binary stabilizer quantum codes which are records according to Grassl (Bounds on the minimum distance of linear codes, http://www.codetables.de, 2020) and new q-ary ones, with q ne 2, improving others in the literature.

Linking Reversed and Dual Codes of Quasi-Cyclic Codes

It is known that quasi-cyclic (QC) codes over the finite field 𝔽q correspond to certain 𝔽q[x]-modules. A QC code C is specified by a generator polynomial matrix G whose rows generate C as an 𝔽q[x]-module. The reversed code of C, denoted by R, is the code obtained by reversing all codewords of C while the dual code of C is denoted by C⊥. We call C reversible, self-orthogonal, and self-dual if R = C, C⊥ ⊇ C, and C⊥ = C, respectively. In this study, for a given C, we find an explicit formula for a generator polynomial matrix of R. A necessary and sufficient condition for C to be reversible is derived from this formula. In addition, we reveal the relations among C, R, and C⊥. Specifically, we give conditions on G corresponding to C⊥ ⊇ R, C⊥ ⊆ R, and C = R = C⊥. As an application, we employ these theoretical results to the construction of QC codes with best parameters. Computer search is used to show that there exist various binary reversible self-orthogonal QC codes that achieve the upper bounds on the minimum distance of linear codes.

Approaching the Minimum Distance Problem by Algebraic Swarm-Based Optimizations

Finding the minimum distance of linear codes is one of the main problems in coding theory. The importance of the minimum distance comes from its error-correcting and error-detecting capability of the handled codes. 
 It was proven that this problem is an NP-hard that is the solution of this problem can be guessed and verified in polynomial time but no particular rule is followed to make the guess and some meta-heuristic approaches in the literature have been used to solve this problem.
 In this paper, swarm-based optimization techniques, bat and firefly, are applied to the minimum distance problem by integrating the algebraic operator to the handled algorithms.

On the Calculation of the Minimum Distance of a Linear Code

Some aspects of the algorithms for calculating the minimum distance of linear codes over finite fields are presented.

Genetic algorithms with permutation-based representation for computing the distance of linear codes

Abstract Finding the minimum distance of linear codes is an NP-hard problem. Traditionally, this computation has been addressed by means of the design of algorithms that find, by a clever exhaustive search, a linear combination of some generating matrix rows that provides a codeword with minimum weight. Therefore, as the dimension of the code or the size of the underlying finite field increase, so it does exponentially the run time. In this work, we prove that, given a generating matrix, there exists a column permutation which leads to a reduced row echelon form containing a row whose weight is the code distance. This result enables the use of permutations as representation scheme, in contrast to the usual discrete representation, which makes the search of the optimum polynomial time dependent from the base field. In particular, we have implemented genetic and CHC algorithms using this representation as a proof of concept. Experimental results have been carried out employing codes over fields with two and eight elements, which suggests that evolutionary algorithms with our proposed permutation encoding are competitive with regard to existing methods in the literature. As a by-product, we have found and amended some inaccuracies in the Magma Computational Algebra System concerning the stored distances of some linear codes.

Self-dual codes over $\mathbb {F}_{2} \times (\mathbb {F}_{2}+v\mathbb {F}_{2})$

In this study we consider Euclidean and Hermitian self-dual codes over the direct product ring $\mathbb {F}_{2} \times (\mathbb {F}_{2}+v\mathbb {F}_{2})$ where v2 = v. We obtain some theoretical outcomes about self-dual codes via the generator matrices of free linear codes over $\mathbb {F}_{2} \times (\mathbb {F}_{2}+v\mathbb {F}_{2})$ . Also, we obtain upper bounds on the minimum distance of linear codes for both the Lee distance and the Gray distance. Moreover, we find some free Euclidean and free Hermitian self-dual codes over $\mathbb {F}_{2} \times (\mathbb {F}_{2}+v\mathbb {F}_{2})$ via some useful construction methods.

New Efficient Scheme Based on Reduction of the Dimension in the Multiple Impulse Method to Find the Minimum Distance of Linear Codes

In order to find a minimum weight codeword in a linear code, the Multiple Impulse Method uses the Ordered Statistics Decoder of order 3 having a complexity which increases with the code dimension. This paper presents an important improvement of this method by finding a sub code of C of small dimension containing a lowest weight codeword. In the case of Binary Extended Quadratic Residue codes, the proposed technique consists on finding a self invertible permutation ( from the projective special linear group and searching a codeword having the minimum weight in the sub code fixed by (. The proposed technique gives the exact value of the minimum distance for all binary quadratic residue codes of length less than 223 by using the Multiple Impulse Method on the sub codes in less than one second. For lengths more than 223, the obtained results prove the height capacity of the proposed technique to find the lowest weight in less time. The proposed idea is generalized for BCH codes and it has permits to find the true value of the minimum distance for some codes of lengths 1023 and 2047. The proposed methods performed very well in comparison to previously known results.

Minimum distance of linear codes and the α-invariant

The simple interpretation of the minimum distance of a linear code obtained by De Boer and Pellikaan, and later refined by the second author, is further developed through the study of various finitely generated graded modules. We use the methods of commutative/homological algebra to find connections between the minimum distance and the α-invariant of such modules.

A Note on the Rank Bounded Distance and Its Algorithms for Cyclic Codes

The minimum distance for linear codes is one of the important parameters. The shift bound is a good lower bound of the minimum distance for cyclic codes, Reed-Muller codes and geometric Goppa codes. It is necessary to construct the maximum value of the independent set for the calculation of the shift bound. However, its computational complexity is very large, because the construction of the independent set is not unique. The authors proposed an algorithm for calculation of the independent set using the discrete Fourier transform in 2010. In this paper we give simple modification and new recurrent algorithms to improve the original algorithm.

Classification and nonexistence results for linear codes with prescribed minimum distances

Starting from a linear [n, k, d] q code with dual distance $${d^{\bot}}$$ , we may construct an $${[n - d^\bot, k - d^\bot +1,\geq d]_q}$$ code with dual distance at least $${\left\lceil\frac{d^\bot}{q}\right\rceil}$$ using construction Y 1. The inverse construction gives a rule for the classification of all [n, k, d] q codes with dual distance $${d^{\bot}}$$ by adding $${d^\bot}$$ further columns to the parity check matrices of the smaller codes. Isomorph rejection is applied to guarantee a small search space for this iterative approach. Performing a complete search based on this observation, we are able to prove the nonexistence of linear codes for 16 open parameter sets [n, k, d] q , q = 2, 3, 4, 5, 7, 8. These results imply 217 new upper bounds in the known tables for the minimum distance of linear codes and establish the exact value in 109 cases.

On the Computing of the Minimum Distance of Linear Block Codes by Heuristic Methods

The evaluation of the minimum distance of linear block codes remains an open problem in coding theory, and it is not easy to determine its true value by classical methods, for this reason the problem has been solved in the literature with heuristic techniques such as genetic algorithms and local search algorithms. In this paper we propose two approaches to attack the hardness of this problem. The first approach is based on genetic algorithms and it yield to good results comparing to another work based also on genetic algorithms. The second approach is based on a new randomized algorithm which we call "Multiple Impulse Method (MIM)", where the principle is to search codewords locally around the all-zero codeword perturbed by a minimum level of noise, anticipating that the resultant nearest nonzero codewords will most likely contain the minimum Hamming-weight codeword whose Hamming weight is equal to the minimum distance of the linear code.

QUASI-CYCLIC CODES FROM CYCLIC-STRUCTURED DESIGNS WITH GOOD PROPERTIES

In this paper, one-generator binary quasi-cyclic (QC) codes are explored by statistical tools derived from design of experiments. A connection between a structured cyclic class of statistical designs, k-circulant supersaturated designs and QC codes is given. The mathematical structure of the later codes is explored and a link between complementary dual binary QC codes and E(s2)-optimal k-circulant supersaturated designs is established. Moreover, binary QC codes of rate 1/3, 1/4, 1/5, 1/6 and 1/7 are found by utilizing a genetic algorithm. Our approach is based on a search for good or best codes that attain the current best-known lower bounds on the minimum distance of linear codes, formulated as a combinatorial optimization problem. Surveying previous results, it is shown, that our codes reach the current best-known lower bounds on the minimum distance of linear codes with the same parameters.

On computing the minimum distance of linear codes

On computing the minimum distance of linear codes

Bounds on Minimum Distance for Linear Codes over GF(5)

Let [n, k, d; q]-codes be linear codes of length n, dimension k and minimum Hamming distance d over GF(q). Let d 5(n, k) be the maximum possible minimum Hamming distance of a linear [n, k, d; 5]-code for given values of n and k. In this paper, forty four new linear codes over GF(5) are constructed and a table of d 5(n, k) k≤ 8, n≤ 100 is presented.

The minimum distance of codes in an array coming from telescopic semigroups

The concept of an error-correcting array gives a new bound on the minimum distance of linear codes and a decoding algorithm which decodes up to half this bound. This gives a unified point of view which explains several improvements on the minimum distance of algebraic-geometric codes. Moreover, it is explained in terms of linear algebra and the theory of semigroups only.