Minimum Weight Codewords Research Articles

On the minimum distance, minimum weight codewords, and the dimension of projective Reed-Muller codes

On the minimum distance, minimum weight codewords, and the dimension of projective Reed-Muller codes

Efficient Algorithms for Constructing Minimum-Weight Codewords in Some Extended Binary BCH Codes

Efficient Algorithms for Constructing Minimum-Weight Codewords in Some Extended Binary BCH Codes

New examples of self-dual near-extremal ternary codes of length 48 derived from 2-[formula omitted] designs

In a recent paper (Araya and Harada, 2023), Araya and Harada gave examples of self-dual near-extremal ternary codes of length 48 for 145 distinct values of the number A12 of codewords of minimum weight 12, and raised the question about the existence of codes for other values of A12. In this note, we use symmetric 2-(47,23,11) designs with an automorphism group of order 6 to construct self-dual near-extremal ternary codes of length 48 for 150 new values of A12.

Dynamic Frozen-Function Design for Reed-Muller Codes With Automorphism-Based Decoding

In this letter, we propose to add dynamic frozen bits to underlying polar codes with a Reed-Muller information set with the aim of maintaining the same sub-decoding structure in Automorphism Ensemble (AE) and lowering the Maximum Likelihood (ML) bound by reducing the number of minimum weight codewords. We provide the dynamic freezing constraint matrix that remains identical after applying a permutation linear transformation. This feature also permits to drastically reduce the memory requirements of an AE decoder with polar-like codes having dynamic frozen bits. We show that, under AE decoding, the proposed dynamic freezing constraints lead to a gain of up to 0.25dB compared to the ML bound of the R(3,7) Reed-Muller code, at the cost of small increase in memory requirements.

On the Minimum Weight Codewords of PAC Codes: The Impact of Pre-Transformation

On the Minimum Weight Codewords of PAC Codes: The Impact of Pre-Transformation

An infinite family of antiprimitive cyclic codes supporting Steiner systems $$S(3,8, 7^m+1)$$

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 few 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 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.

On some projective triply-even binary codes invariant under the Conway group ${\rm Co}_1$

A binary triply-even $[98280, 25, 47104]_2$ code invariant under the sporadic simple group ${rm Co}_1$ is constructed by adjoining the all-ones vector to the faithful and absolutely irreducible 24-dimensional code of length 98280. Using the action of ${rm Co}_1$ on the code we give a description of the nature of the codewords of any non-zero weight relating these to vectors of types 2, 3 and 4, respectively of the Leech lattice. We show that the stabilizer of any non-zero weight codeword in the code is a maximal subgroup of ${rm Co}_1$. Moreover, we give a partial description of the nature of the codewords of minimum weight of the dual code.

Majority Logic Decoding for Certain Schubert Codes Using Lines in Schubert Varieties

In this article, we consider Schubert codes, linear codes associated to Schubert varieties, and discuss minimum weight codewords for dual Schubert codes. The notion of lines in Schubert varieties is looked closely at, and it has been proved that the supports of the minimum weight codewords of the dual Schubert codes lie on lines and any three points on a line in Schubert variety correspond to the support of some minimum weight parity check for the Schubert code. We use these lines in Schubert varieties to construct orthogonal parity checks for certain Schubert codes and use them for majority logic decoding. In some special cases, we can correct approximately up to $\lfloor (d-1)/2\rfloor$ many errors where $d$ is the minimum distance of the code.

Linear codes of 2-designs as subcodes of the generalized Reed-Muller codes

This paper is devoted to the affine-invariant ternary codes defined by Hermitian functions. We first compute the incidence matrices of the 2-designs supported by the minimum weight codewords of these ternary codes. Then we show that the linear codes spanned by the rows of these incidence matrices are subcodes of the 4-th order generalized Reed-Muller codes and also hold 2-designs. Finally, we determine the dimension and develop a lower bound on the minimum distance of the ternary linear codes.

Small weight codewords of projective geometric codes

We investigate small weight codewords of the p-ary linear code Cj,k(n,q) generated by the incidence matrix of k-spaces and j-spaces of PG(n,q) and its dual, with q a prime power and 0⩽j<k<n. Firstly, we prove that all codewords of Cj,k(n,q) up to weight (3−O(1q))[k+1j+1]q are linear combinations of at most two k-spaces (i.e. two rows of the incidence matrix). As for the dual code Cj,k(n,q)⊥, we manage to reduce both problems of determining its minimum weight (1) and characterising its minimum weight codewords (2) to the case C0,1(n,q)⊥. This implies the solution to both problem (1) and (2) if q is prime and the solution to problem (1) if q is even.

On the weights of dual codes arising from the GK curve

In this paper we investigate some dual algebraic-geometric codes associated with the Giulietti–Korchmáros maximal curve. We compute the minimum distance and the minimum weight codewords of such codes and we investigate the generalized Hamming weights of such codes.

Efficacy of the Metropolis Algorithm for the Minimum-Weight Codeword Problem Using Codeword and Generator Search Spaces

This article studies the efficacy of the Metropolis algorithm for the minimum-weight codeword problem . The input is a linear code $C$ given by its generator matrix and our task is to compute a nonzero codeword in the code $C$ of least weight. In particular, we study the Metropolis algorithm on two possible search spaces for the problem: 1) the codeword space and 2) the generator space . The former is the space of all codewords of the input code and is the most natural one to use and hence has been used in previous work on this problem. The latter is the space of all generator matrices of the input code and is studied for the first time in this article. In this article, we show that for an appropriately chosen temperature parameter the Metropolis algorithm mixes rapidly when either of the search spaces mentioned above are used. Experimentally, we demonstrate that the Metropolis algorithm performs favorably when compared to previous attempts. When using the generator space, the Metropolis algorithm is able to outperform the previous algorithms in most of the cases. We have also provided both theoretical and experimental justification to show why the generator space is a worthwhile search space to use for this problem.

Locally repairable codes with high availability based on generalised quadrangles

&lt;p style='text-indent:20px;'&gt;Locally Repairable Codes (LRC's) based on generalised quadrangles were introduced by Pamies-Juarez, Hollmann and Oggier in [&lt;xref ref-type="bibr" rid="b3"&gt;3&lt;/xref&gt;], and bounds on the repairability and availability were derived. In this paper, we determine the values of the repairability and availability of such LRC's for a large portion of the currently known generalised quadrangles. In order to do so, we determine the minimum weight of the codes of translation generalised quadrangles and characterise the codewords of minimum weight.&lt;/p&gt;

ECC2: Error correcting code and elliptic curve based cryptosystem

Code-based cryptography has aroused wide public concern as one of the main candidates for post quantum cryptography to resist attacks against cryptosystems from quantum computation. However, the large key size becomes a drawback that prevents it from wide practical applications although it performs pretty well on the speed of both encryption and decryption. The use of algebraic geometry codes is considered to be a good solution to reduce the key size, but the special structures of algebraic geometry codes results in lots of attacks including Minder’s attack. To cope with the barriers of large key size as well as attacks from the special structures of algebraic codes, we propose a code-based encryption system using elliptic codes. The special structure of elliptic codes helps us to effectively reduce the size of secret key. By choosing the rational points carefully, we build elliptic codes whose minimum weight codeword is hard to sample. Such codes are used in constructing encryption systems such that Minder’s attacks can be resisted. More importantly, we apply the list decoding algorithm in the decryption process thus more errors beyond half of the minimum distance of the code could be corrected, which is the key point to resist other known attacks for algebraic geometry codes based cryptosystems. Our implementation shows that the proposed encryption system performs well on the key size and ciphertext expansion rate.

Linear representations of finite geometries and associated LDPC codes

The linear representation of a subset of a finite projective space is an incidence system of affine points and lines determined by the subset. In this paper we use character theory to show that the rank of the incidence matrix has a direct geometric interpretation in terms of certain hyperplanes. We consider the LDPC codes defined by taking the incidence matrix and its transpose as parity-check matrices, and in the former case prove a conjecture of Vandendriessche that the code is generated by words of minimum weight called plane words. In the latter case we compute the minimum weight in several cases and provide explicit constructions of minimum weight codewords.

Linear codes of 2-designs associated with subcodes of the ternary generalized Reed–Muller codes

In this paper, the 3-rank of the incidence matrices of 2-designs supported by the minimum weight codewords in a family of ternary linear codes considered in [C. Ding, C. Li, Infinite families of 2-designs and 3-designs from linear codes, Discrete Mathematics 340(10) (2017) 2415--2431] are computed. A lower bound on the minimum distance of the ternary codes spanned by the incidence matrices of these designs is derived, and it is proved that the codes are subcodes of the 4th order generalized Reed-Muller codes.

Hermitian codes and complete intersections

In this paper we consider the Hermitian codes defined as the dual codes of one-point evaluation codes on the Hermitian curve H over the finite field Fq2. We focus on those with distance d≥q2−q and give a geometric description of the support of their minimum-weight codewords. We consider the unique writing μq+λ(q+1) of the distance d with μ,λ non negative integers, and μ≤q, and consider all the curves X of the affine plane AFq22 of degree μ+λ defined by polynomials with xμyλ as leading monomial with respect to the DegRevLex term ordering (with y>x). We prove that a zero-dimensional subscheme Z of AFq22 is the support of a minimum-weight codeword of the Hermitian code with distance d if and only if it is made of d simple Fq2-points and there is a curve X such that Z coincides with the scheme theoretic intersection H∩X (namely, as a cycle, Z=H⋅X). Finally, exploiting this geometric characterization, we propose an algorithm to compute the number of minimum weight codewords and we present comparison tables between our algorithm and MAGMA command MinimumWords.

Minimum-weight codewords of the Hermitian codes are supported on complete intersections

Let H be the Hermitian curve defined over a finite field Fq2. In this paper we complete the geometrical characterization of the supports of the minimum-weight codewords of the algebraic-geometry codes over H, started in [26]: if d is the distance of the code, the supports are all the sets of d distinct Fq2-points on H complete intersection of two curves defined by polynomials with prescribed initial monomials w.r.t. DegRevLex.For most Hermitian codes, and especially for all those with distance d≥q2−q studied in [26], one of the two curves is always the Hermitian curve H itself, while if d<q the supports are complete intersection of two curves none of which can be H.Finally, for some special codes among those with intermediate distance between q and q2−q, both possibilities occur.We provide simple and explicit numerical criteria that allow to decide for each code what kind of supports its minimum-weight codewords have and to obtain a parametric description of the family (or the two families) of the supports.

Minimum weight codewords in dual algebraic-geometric codes from the Giulietti-Korchmáros curve

In this paper we investigate the number of minimum weight codewords of some dual algebraic-geometric codes associated with the Giulietti–Korchmaros maximal curve, by computing the maximal number of intersections between the Giulietti–Korchmaros curve and lines, plane conics, and plane cubics.

A New Way to Find the Minimum Distance of BCH and Quadratic Double Circulant Codes

BCH (Bose, Chaudhuri, Hocquenghem) and QDC (Quadratic Double Circulant) codes are important classes of linear codes with many application domains. For large lengths, the true value of the minimum distances of these codes remains undetermined and therefore their error correcting capabilities are still unknown. The problem of searching codewords of lowest weight in linear codes is NP-hard problem and it is equivalent to the problem of the minimum distance research. Many methods have been used to attack this hardness such as genetic algorithms, simulated annealing, Multiple Impulse Method (MIM), Chen algorithm, Canteaut-Chabaud and Zimmermann algorithms. In a previous work, we have presented the method MIM-RSC to find the minimum weights by applying the MIM method on some Random sub codes of reduced dimensions. This reduction allows an efficient local search, but the important question here is which are the sub codes containing the lowest weights and how find them?. In general, the answer of this question isn’t evident and consequently many random sub codes should be exploited to have a great chance for taking a true minimum weight codeword. In this work, we present a new way to catch lowest weights by applying the Zimmermann algorithm on these random sub codes. This proposed scheme, Zimmermann-RSC, is validated on some BCH and QDC codes of known minimum weight and it is used to determine the minimum distance for many other codes of large lengths belonging to these families. The comparison between some competitors methods with the proposed scheme Zimmermann-RSC on many large BCH and QDC codes proves the efficiency of this latest in terms of the results quality and run time.