Distance Of Codes Research Articles

A class of t-weight codes and its applications

In this paper, we constructed a class of [Formula: see text]-weight linear codes over [Formula: see text] under the homogeneous weight metric by their generator matrices, where [Formula: see text] and [Formula: see text] The Gray images of some class of these codes over [Formula: see text] are [Formula: see text]-ary nonlinear codes, which have the same weight distributions as that of the two-weight [Formula: see text]-ary linear codes of type SU1 in the sense of [R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. London Math. Soc. 18(2) (1986) 97–122]. Also, we obtained the minimum distance of the dual codes of the constructed codes. Further, we discussed some optimal linear codes over [Formula: see text] with respect to Plotkin-type bound from the constructed codes when [Formula: see text] Furthermore, we investigated the applications in strongly regular graphs and secret sharing schemes.

Some Quaternary Additive Codes Outperform Linear Counterparts

The additive codes may have better parameters than linear codes. However, it is still a challenging problem to efficiently construct additive codes that outperform linear codes, especially those with greater distances than linear codes of the same lengths and dimensions. This paper focuses on constructing additive codes that outperform linear codes based on quasi-cyclic codes and combinatorial methods. Firstly, we propose a lower bound on the symplectic distance of 1-generator quasi-cyclic codes of index even. Secondly, we get many binary quasi-cyclic codes with large symplectic distances utilizing computer-supported combination and search methods, all of which correspond to good quaternary additive codes. Notably, some additive codes have greater distances than best-known quaternary linear codes in Grassl’s code table (bounds on the minimum distance of quaternary linear codes http://www.codetables.de) for the same lengths and dimensions. Moreover, employing a combinatorial approach, we partially determine the parameters of optimal quaternary additive 3.5-dimensional codes with lengths from 28 to 254. Finally, as an extension, we also construct some good additive complementary dual codes with larger distances than the best-known quaternary linear complementary dual codes in the literature.

Construction of a Class of Real Array Rank Distance Codes

Rank distance codes are known to be applicable in various applications such as distributed data storage, cryptography, space time coding, and mainly in network coding. Rank distance codes defined over finite fields have attracted considerable attention in recent years. However, in some scenarios where codes over finite fields are not sufficient, it is demonstrated that codes defined over the real number field are preferred. In this paper, we proposed a new class of rank distance codes over the real number field R . The real array rank distance (RARD) codes we constructed here can be used for all the applications mentioned above whenever the code alphabet is the real field R . From the class of RARD codes, we extract a subclass of equidistant constant rank codes which is applicable in network coding. Also, we determined an upper bound for the dimension of RARD codes leading the way to obtain some optimal RARD codes. Moreover, we established examples of some RARD codes and optimal RARD codes.

On constacyclic codes of length 9ps over 𝔽pm and their optimal codes

The problem of classifying constacyclic codes over a finite field, both the Hamming distance and the algebraic structure, is an interesting problem in algebraic coding theory. For the repeated-root constacyclic codes of length [Formula: see text] over [Formula: see text], where [Formula: see text] is a prime number and [Formula: see text] does not divide [Formula: see text], the problem has been solved completely for all [Formula: see text] and partially for [Formula: see text]. In this paper, we solve the problem for [Formula: see text] and all primes [Formula: see text] different from [Formula: see text] and [Formula: see text]. In particular, we characterize the Hamming distance of all repeated-root constacyclic codes of length [Formula: see text] over [Formula: see text]. As an application, we identify all optimal and near-optimal codes with respect to the Singleton bound of these types, namely, MDS, almost-MDS, and near-MDS codes.

Zeta functions for tensor codes

In this work, we introduce a new class of optimal tensor codes related to the Ravagnani-type anticodes defined in 2023. We show that it extends the family of [Formula: see text]-maximum rank distance codes and contains the [Formula: see text]-binomial moment determined codes (with respect to the Ravagnani-type anticodes) as a proper subclass. We define and study the zeta function for tensor codes. We establish connections between this object and the weight enumerator of a tensor code with respect to the Ravagnani-type anticodes. We introduce a new refinement of the invariants of tensor codes exploiting the structure of product lattices of some classes of anticodes classified in 2023 and we derive the corresponding MacWilliams identities. In this framework, we also define a multivariate version of the tensor weight enumerator and we establish relations with the corresponding zeta function. As an application, we derive connections on the tensor weights.

Flexible coding of time or distance in hippocampal cells.

Analysis of neuronal activity in the hippocampus of behaving animals has revealed cells acting as 'Time Cells', which exhibit selective spiking patterns at specific time intervals since a triggering event, and 'Distance Cells', which encode the traversal of specific distances. Other neurons exhibit a combination of these features, alongside place selectivity. This study aims to investigate how the task performed by animals during recording sessions influences the formation of these representations. We analyzed data from a treadmill running study conducted by Kraus et al.1 in which rats were trained to run at different velocities. The rats were recorded in two trial contexts: a 'fixed time' condition, where the animal ran on the treadmill for a predetermined duration before proceeding, and a 'fixed distance' condition, where the animal ran a specific distance on the treadmill. Our findings indicate that the type of experimental condition significantly influenced the encoding of hippocampal cells. Specifically, distance-encoding cells dominated in fixed-distance experiments, whereas time-encoding cells dominated in fixed-time experiments. These results underscore the flexible coding capabilities of the hippocampus, which are shaped by over-representation of salient variables associated with reward conditions.

Proximity Gaps for Reed–Solomon Codes

A collection of sets displays a proximity gap with respect to some property if for every set in the collection, either (i) all members are δ-close to the property in relative Hamming distance or (ii) only a tiny fraction of members are δ-close to the property. In particular, no set in the collection has roughly half of its members δ-close to the property and the others δ-far from it. We show that the collection of affine spaces displays a proximity gap with respect to Reed–Solomon (RS) codes, even over small fields, of size polynomial in the dimension of the code, and the gap applies to any δ smaller than the Johnson/Guruswami–Sudan list-decoding bound of the RS code. We also show near-optimal gap results, over fields of (at least) linear size in the RS code dimension, for δ smaller than the unique decoding radius. Concretely, if δ is smaller than half the minimal distance of an RS code V ⊂ 𝔽 q n , then every affine space is either entirely δ-close to the code or, alternatively, at most an ( n/q )-fraction of it is δ-close to the code. Finally, we discuss several applications of our proximity gap results to distributed storage, multi-party cryptographic protocols, and concretely efficient proof systems. We prove the proximity gap results by analyzing the execution of classical algebraic decoding algorithms for Reed–Solomon codes (due to Berlekamp–Welch and Guruswami–Sudan) on a formal element of an affine space. This involves working with Reed–Solomon codes whose base field is an (infinite) rational function field. Our proofs are obtained by developing an extension (to function fields) of a strategy of Arora and Sudan for analyzing low-degree tests.

Evaluation of Skeletonization Techniques for 2D Binary Images

In the realm of modern image processing, the emphasis often lies on engineering-based approaches rather than scientific solutions to address diverse practical problems. One prevalent task within this domain involves the skeletonization of binary images. Skeletonization is a powerful process for extracting the skeleton of objects located in digital binary images. This process is widely employed for automating many tasks in numerous fields such as pattern recognition, robot vision, animation, and image analysis. The existing skeletonization techniques are mainly based on three approaches: boundary erosion, distance coding, and Voronoi diagram for identifying an approximate skeleton. In this work, we present an empirical evaluation of a set of well-known techniques and report our findings. We specifically deal with computing skeletons in 2d binary images by selecting different approaches and evaluating their effectiveness. Visual evaluation is the primary method used to showcase the performance of selected skeletonization algorithms. Due to the absence of a definitive definition for the "true" skeleton of a digital object, accurately assessing the effectiveness of skeletonization algorithms poses a significant research challenge. Although researchers have attempted quantitative assessments, these measures are typically customized for specific domains and may not be suitable for our current work. The experimental results shown in this work illustrate the performance of the three main approaches in applying skeletonization with respect to different perspectives.

On the equivalence issue of a class of 2-dimensional linear maximum rank distance codes

In Neri et al. (2022) [20], the authors extended the family of 2-dimensional Fq2t-linear MRD codes recently found in Longobardi et al. (2023) [13]. Also, for t≥5 they determined equivalence classes of the elements in this new family and provided the exact number of inequivalent codes in it. In this article, we complete the study of the equivalence issue removing the restriction t≥5. Moreover, we prove that in the case when t=4, the linear sets of the projective line PG(1,q8) ensuing from codes in the relevant family, are not equivalent to any one known so far.

Convolutional-Neural-Network-Based Hexagonal Quantum Error Correction Decoder

Topological quantum error-correcting codes are an important tool for realizing fault-tolerant quantum computers. Heavy hexagonal coding is a new class of quantum error-correcting coding that assigns physical and auxiliary qubits to the vertices and edges of a low-degree graph. The layout of heavy hexagonal codes is particularly suitable for superconducting qubit architectures to reduce frequency conflicts and crosstalk. Although various topological code decoders have been proposed, constructing the optimal decoder remains challenging. Machine learning is an effective decoding scheme for topological codes, and in this paper, we propose a machine learning heavy hexagonal decoder based on a convolutional neural network (CNN) to obtain the decoding threshold. We test our method on heavy hexagonal codes with code distance of three, five, and seven, and increase it to five, seven, and nine by optimizing the RestNet network architecture. Our results show that the decoder thresholding accuracies are about 0.57% and 0.65%, respectively, which are about 25% higher than the conventional decoding scheme under the depolarizing noise model. The proposed decoding architecture is also applicable to other topological code families.

On Hamming distance distributions of repeated-root constacyclic codes of length 3ps over [formula omitted

Let p≠3 be any prime. The algebraic structures of all λ-constacyclic codes of length 3ps over the finite commutative chain ring R=Fpm+uFpm are provided by classifying all ideals of the local ring R[x]〈x3ps−λ〉. Using these structures, in this paper, the exact values of Hamming distances of all such λ-constacyclic codes of length 3ps are established. As an application, we identify all maximal distance separable λ-constacyclic codes of length 3ps.

Context-Aware Expert for Software Architecture Recovery (CAESAR): An automated approach for recovering software architectures

Software architecture recovery approaches help to reconstruct the architecture of complex software systems. However, the manual effort and expertise required to use such techniques are high, as the user demands detailed knowledge about software architecture recovery process and preparing additional software artifacts to use as input for such approaches. Such practices lack consideration for the context standards of the recovered software. Additionally, proper module detection rules that are aware of the modeled system’s context should be studied to enhance the cluster quality of the detected modules. We introduce a novel approach to software architecture recovery by implementing an automated knowledge-based rules solution that considers the context of the developed software while identifying and clustering that system’s software modules. Such solution is aided by a novel code distance model that is utilized to enhance the cohesiveness of the recovered software modules, improving the recovered software architecture quality, and enhancing readability through providing visualizations at different abstraction levels for the recovered architecture. The proposed method has been evaluated on two software systems with varying sizes and from different contexts and involved experienced participants. The results demonstrate significant improvements in cohesion and coupling of the produced clusters compared to alternative architecture recovery approaches.

Adaptive syndrome measurements for Shor-style error correction

The Shor fault-tolerant error correction (FTEC) scheme uses transversal gates and ancilla qubits prepared in the cat state in syndrome extraction circuits to prevent propagation of errors caused by gate faults. For a stabilizer code of distance d that can correct up to t=⌊(d−1)/2⌋ errors, the traditional Shor scheme handles ancilla preparation and measurement faults by performing syndrome measurements until the syndromes are repeated t+1 times in a row; in the worst-case scenario, (t+1)2 rounds of measurements are required. In this work, we improve the Shor FTEC scheme using an adaptive syndrome measurement technique. The syndrome for error correction is determined based on information from the differences of syndromes obtained from consecutive rounds. Our protocols that satisfy the strong and the weak FTEC conditions require no more than (t+3)2/4−1 rounds and (t+3)2/4−2 rounds, respectively, and are applicable to any stabilizer code. Our simulations of FTEC protocols with the adaptive schemes on hexagonal color codes of small distances verify that our protocols preserve the code distance, can increase the pseudothreshold, and can decrease the average number of rounds compared to the traditional Shor scheme. We also find that for the code of distance d, our FTEC protocols with the adaptive schemes require no more than d rounds on average.

Decoding Quantum Tanner Codes

We introduce sequential and parallel decoders for quantum Tanner codes. When the Tanner code construction is applied to a sufficiently expanding square complex with robust local codes, we obtain a family of asymptotically good quantum low-density parity-check codes. In this case, our decoders provably correct arbitrary errors of weight linear in the code length, respectively in linear or logarithmic time. The same decoders are easily adapted to the expander lifted product codes of Panteleev and Kalachev. Along the way, we exploit recently established bounds on the robustness of random tensor codes to give a tighter bound on the minimum distance of quantum Tanner codes.

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

Cones from maximum h-scattered linear sets and a stability result for cylinders from hyperovals

This paper mainly focuses on cones whose basis is a maximum h-scattered linear set. We start by investigating the intersection numbers of such cones with respect to the hyperplanes. Then we analyze two constructions of point sets with few intersection numbers with respect to the hyperplanes. In particular, the second one extends the construction of translation KM-arcs in projective spaces, having as part at infinity a cone with basis a maximum h-scattered linear set. As an instance of the second construction we obtain cylinders with a hyperoval as basis, which we call hypercylinders, for which we are able to provide a stability result. The main motivation for these problems is related to the connections with both Hamming and rank distance codes. Indeed, we are able to construct codes with few weights and to provide a stability result for the codes associated with hypercylinders.

A scalable and fast artificial neural network syndrome decoder for surface codes

Surface code error correction offers a highly promising pathway to achieve scalable fault-tolerant quantum computing. When operated as stabilizer codes, surface code computations consist of a syndrome decoding step where measured stabilizer operators are used to determine appropriate corrections for errors in physical qubits. Decoding algorithms have undergone substantial development, with recent work incorporating machine learning (ML) techniques. Despite promising initial results, the ML-based syndrome decoders are still limited to small scale demonstrations with low latency and are incapable of handling surface codes with boundary conditions and various shapes needed for lattice surgery and braiding. Here, we report the development of an artificial neural network (ANN) based scalable and fast syndrome decoder capable of decoding surface codes of arbitrary shape and size with data qubits suffering from the depolarizing error model. Based on rigorous training over 50 million random quantum error instances, our ANN decoder is shown to work with code distances exceeding 1000 (more than 4 million physical qubits), which is the largest ML-based decoder demonstration to-date. The established ANN decoder demonstrates an execution time in principle independent of code distance, implying that its implementation on dedicated hardware could potentially offer surface code decoding times of O(μsec), commensurate with the experimentally realisable qubit coherence times. With the anticipated scale-up of quantum processors within the next decade, their augmentation with a fast and scalable syndrome decoder such as developed in our work is expected to play a decisive role towards experimental implementation of fault-tolerant quantum information processing.

Two Conjectures on the Largest Minimum Distances of Binary Self-Orthogonal Codes With Dimension 5

The purpose of this paper is to solve the two conjectures on the largest minimum distance <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d_so</i> ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> , 5) of a binary self-orthogonal [ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> , 5] code proposed by Kim and Choi (2022). The determination of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d_so</i> ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n, k</i> ) has been a fundamental and difficult problem in coding theory because there are too many binary self-orthogonal codes as the dimension <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> increases. Recently, Kim et al. (2021) considered the shortest self-orthogonal embedding of a binary linear code, and many binary optimal self-orthogonal [ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n, k</i> ] codes were constructed for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> = 4, 5. Kim and Choi (2022) improved some results of Kim et al. (2021) and made two conjectures on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d_so</i> ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> , 5). In this paper, we develop a general method to determine the exact value of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d_so</i> ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n, k</i> ) for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> = 5, 6 and show that the two conjectures made by Kim and Choi (2022) are true.

Распределение мощности кодов с наименьшей избыточностью алфавитов в зависимости от количества бит и кодового расстояния

Purpose: To investigate the dependence of the maximum power of codes on the number of digits and the minimum code distance; to find an approach to determine the optimal rules for constructing the check vector of a separable code from the point of view of ensuring minimal redundancy with a given reliability of message transmission. Methods: Computer simulation has been used to conduct experimental studies. For theoretical studies, the method of analytical review, graph theory, and coding theory have been applied. Results: Theoretical and experimental studies have obtained certain specific cases of power distribution for code alphabets with a given Hamming distance for various constant lengths, generated using the previously described algorithm. A method for doubling the power of arbitrary binary codes is proposed and described, as well as a method for obtaining codes with the least redundancy of powers M=2f, where f is a natural number for a predetermined minimum code distance by recursively using the method proposed in the article. Practical significance: An algorithm has been developed for doubling the power of the code alphabet while maintaining the required reliability of data transmission. A technique for analyzing the resulting matrices of code vectors is obtained in order to determine the rules for calculating check bits without using cyclic algorithms.

Butson–Hadamard matrices and Plotkin-optimal codes over ℤpe

In this paper, we deal with codes obtained from Butson–Hadamard matrices, called BH codes, focusing on their minimum distances. We first consider the usual Hamming distance and find lower bounds for distances of BH codes. Then we turn our attention to homogeneous weights, and search for distances of BH code families under these weights. Next, we introduce the notion of quasi-homogeneous weights as a generalization of homogeneous weights and show that certain BH codes equipped with quasi-homogeneous weights are Plotkin optimal. In addition, we obtain distances of BH codes under certain quasi-homogeneous weights. Our results are applied to determine parameters of [Formula: see text]-ary codes projected under Gray isometries from BH codes over [Formula: see text], where [Formula: see text] is a prime number and [Formula: see text] is an integer.