Maximal Codes Research Articles

Generating efficient circulant-like MDS matrices for implementation

The exploration of maximal distance separable codes (MDS codes) has been a longstanding focus in error-correcting code theory and holds significant relevance in cryptography. Numerous approaches have been investigated for constructing MDS matrices, including deriving them from MDS codes, utilizing Hadamard matrices, Cauchy matrices, Vandermonde matrices, circulant matrices, circulant-like matrices, among others. However, a major challenge for cryptography designers is finding MDS matrices with low implementation cost. In this paper, we propose algorithms for generating efficient circulant-like MDS matrices of size , and for implementation. Subsequently, we evaluate the fixed points, the number of XOR operations of the proposed MDS matrices, and compare them with MDS matrices of other well-known ciphers. These proposed MDS matrices can become promising candidates for many cryptographic algorithms in the future.

Maximal Genetic Code Symmetry Is a Physicochemical Purine-Pyrimidine Symmetry Language for Transcription and Translation in the Flow of Genetic Information from DNA to Proteins.

Until now, research has not taken into consideration the physicochemical purine-pyrimidine symmetries of the genetic code in the transcription and translation processes of proteinogenesis. Our Supersymmetry Genetic Code table, developed in 2022, is common and unique for all RNA and DNA living species. Its basic structure is a purine-pyrimidine symmetry net with double mirror symmetry. Accordingly, the symmetry of the genetic code directly shows its organisation based on the principle of nucleotide Watson-Crick and codon-anticodon pairing. The maximal purine-pyrimidine symmetries of codons show that each codon has a strictly defined and unchangeable position within the genetic code. We discovered that the physicochemical symmetries of the genetic code play a fundamental role in recognising and differentiating codons from mRNA and the anticodon tRNA and aminoacyl-tRNA synthetases in the transcription and translation processes. These symmetries also support the wobble hypothesis with non-Watson-Crick pairing interactions between the translation process from mRNA to tRNA. The Supersymmetry Genetic Code table shows a specific arrangement of the second base of codons, according to which it is possible that an anticodon from tRNA recognises whether a codon from mRNA belongs to an amino acid with two or four codons, which is very important in the purposeful use of the wobble pairing process. Therefore, we show that canonical and wobble pairings essentially do not lead to misreading and errors during translation, and we point out the role of physicochemical purine-pyrimidine symmetries in decreasing disorder according to error minimisation and preserving the integrity of biological processes during proteinogenesis.

Circular code identified by the codon usage

Since 1996, circular codes in genes have been identified thanks to the development of 6 statistical approaches: trinucleotide frequencies per frame (Arquès and Michel, 1996), correlation functions per frame (Arquès and Michel, 1997), frame permuted trinucleotide frequencies (Frey and Michel, 2003, 2006), advanced statistical functions at the gene population level (Michel, 2015) and at the gene level (Michel, 2017). All these 3-frame statistical methods analyse the trinucleotide information in the 3 frames of genes: the reading frame and the 2 shifted frames. Notably, codon usage does not allow for the identification of circular codes (Michel, 2020). This has been a long-standing problem since 1996, hindering biologists’ access to circular code theory.By considering circular code conditions resulting from code theory, particularly the concept of permutation class, and building upon previous statistical work, a new statistical approach based solely on the codon usage, i.e. a 1-frame statistical method, surprisingly reveals the maximal C3 self-complementary trinucleotide circular code X in bacterial genes and in average (bacterial, archaeal, eukaryotic) genes, and almost in archaeal genes. Additionally, a new parameter definition indicates that bacterial and archaeal genes exhibit codon usage dispersion of the same order of magnitude, but significantly higher than that observed in eukaryotic genes. This statistical finding may explain the greater variability of codes in eukaryotic genes compared to bacterial and archaeal genes, an issue that has been open for many years. Finally, biologists can now search for new (variant) circular codes at both the genome level (across all genes in a given genome) and the gene level using only codon usage, without the need for analysing the shifted frames.

Circular cut codes in genetic information

In this work we present an analysis of the dinucleotide occurrences in the three codon sites 1–2, 2–3 and 1–3, based on a computation of the codon usage of three large sets of bacterial, archaeal and eukaryotic genes using the same method that identified a maximal C3 self-complementary trinucleotide circular code X in genes of bacteria and eukaryotes in 1996 (Arquès and Michel, 1996). Surprisingly, two dinucleotide circular codes are identified in the codon sites 1–2 and 2–3. Furthermore, these two codes are shifted versions of each other. Moreover, the dinucleotide code in the codon site 1–3 is circular, self-complementary and contained in the projection of X onto the 1st and 3rd bases, i.e. by cutting the middle base in each codon of X. We prove several results showing that the circularity and the self-complementarity of trinucleotide codes is induced by the circularity and the self-complementarity of its dinucleotide cut codes. Finally, we present several evolutionary approaches for an emergence of trinucleotide codes from dinucleotide codes.

On the size of maximal binary codes with 2, 3, and 4 distances

On the size of maximal binary codes with 2, 3, and 4 distances

Locally maximal recoverable codes and LMR-LCD codes

Locally maximal recoverable codes and LMR-LCD codes

Circular code in introns

A massive statistical analysis based on the autocorrelation function of the circular code X observed in genes is performed on the (eukaryotic) introns. Surprisingly, a circular code periodicity 0 modulo 3 is identified in 5 groups of introns: birds, ascomycetes, basidiomycetes, green algae and land plants. This circular code periodicity, which is a property of retrieving the reading frame in (protein coding) genes, may suggest that these introns have a coding property. In a well-known way, a periodicity 1 modulo 2 is observed in 6 groups of introns: amphibians, fishes, mammals, other animals, reptiles and apicomplexans. A mixed periodicity modulo 2 and 3 is found in the introns of insects. Astonishing, a subperiodicity 3 modulo 6 is a common statistical property in these 3 classes of introns. When the particular trinucleotides N1N2N1 of the circular code X are not considered, the circular code periodicity 0 modulo 3, hidden by the periodicity 1 modulo 2, is now retrieved in 5 groups of introns: amphibians, fishes, other animals, reptiles and insects. Thus, 10 groups of introns, taxonomically different, out of 12 have a coding property related to the reading frame retrieval. The trinucleotides N1N2N1 are analysed in the 216 maximal C3 self-complementary trinucleotide circular codes. A hexanucleotide code (words of 6 letters) is proposed to explain the periodicity 3 modulo 6. It could be a trace of more general circular codes at the origin of the circular code X.

Labeling-Based Recipient Identification with Low-Order Modulation

Labeling-Based Recipient Identification (LABRID) brings the possibility of representing the destination station address in a Bit-Interleaved Coded Modulation with Iterative Decoding (BICM-ID) system by a signal labeling rule. Low-order modulations, such as BPSK or QPSK, pose a general problem for BICM-ID due to a limited convergence of iterative decoding. In the context of LABRID, they have one more drawback—a small number of different labeling rules in general; the number of the optimal ones, which exhibit the maximal asymptotic coding gain, is reduced even further. Meanwhile, LABRID needs a sizable collection of different optimal labeling rules to serve many users in large wireless networks. In this paper, the author suggests the use of hypercube BPSK or QPSK labeling to overcome all these challenges. By means of the Reactive Tabu Search (RTS) algorithm, more than 1500 equivalent optimal hypercube labeling rules are found. Analytical error bounds of the system are developed and supported by simulation experiments. Then, the focus is moved to the criterion to determine the frame destination at the LABRID receiver; a simple threshold-based method is proposed to keep the incorrect decision probability below 10−5. Finally, it is shown that LABRID outperforms a reference BICM-ID system in terms of computational complexity.

State complexity of binary coded regular languages

For the given non-unary input alphabet Σ, a maximal prefix code h mapping strings over Σ to binary strings, and an optimal deterministic finite automaton (DFA) A with n states recognizing a language L over Σ, we consider the problem of how many states we need for an automaton A′ that decides membership in h(L), the binary coded version of L. Namely, A′ accepts binary inputs belonging to h(L) and rejects binary inputs belonging to h(LC), where LC is the complement of L. The outcome on inputs that are not valid binary codes for any string in Σ⁎ can be arbitrary: A′ may accept, reject, or halt in a “don't care” state. We show that every optimal deterministic don't care finite automaton (dcDFA) A′ solving this promise problem uses at most (‖Σ‖−1)⋅n states but at least n states. In addition, we have established a kind of pseudo-isomorphism between such DFAs and dcDFAs. We also show that, for each non-unary input alphabet Σ, there exists a maximal binary prefix code h such that, for each n≥2 and for each N in range from n to (‖Σ‖−1)⋅n, there exists a language L over Σ such that the optimal DFA recognizing L uses exactly n states and every optimal dcDFA for solving the above promise problem uses exactly N states. Thus, we have the complete state hierarchy for deciding membership in the binary coded version of L, with no magic numbers in between the lower and upper bounds.

In search of maximum non-overlapping codes

AbstractNon-overlapping codes are block codes that have arisen in diverse contexts of computer science and biology. Applications typically require finding non-overlapping codes with large cardinalities, but the maximum size of non-overlapping codes has been determined only for cases where the codeword length divides the size of the alphabet, and for codes with codewords of length two or three. For all other alphabet sizes and codeword lengths no computationally feasible way to identify non-overlapping codes that attain the maximum size has been found to date. Herein we characterize maximal non-overlapping codes. We formulate the maximum non-overlapping code problem as an integer optimization problem and determine necessary conditions for optimality of a non-overlapping code. Moreover, we solve several instances of the optimization problem to show that the hitherto known constructions do not generate the optimal codes for many alphabet sizes and codeword lengths. We also evaluate the number of distinct maximum non-overlapping codes.

Triple disentangled network with dual attention for remote sensing image fusion

Remote sensing applications, such as detection and recognition of objects, need a large number of high spatial resolution (HR) images. As a feasible technique, image fusion is considered to produce HR images. However, most of remote sensing image fusion methods based on deep neural network (DNN) are mainly limited by the extraction of spatial and spectral features. These methods ignore the redundancy among these features, which causes spatial and spectral distortions in the fused image. In this paper, we propose a novel image fusion method based on a triple disentangled network (TDNet) with dual attention to reduce the redundancy in the extracted features. In the proposed method, it is assumed that the information in panchromatic (PAN) and low spatial resolution multispectral (LRMS) images can be encoded as the spatial, spectral, and common features. Specifically, we construct a triple-stream network to extract these features. To efficiently model the spatial and spectral information in PAN and LRMS images, local–global attention and interdependency attention are designed and integrated into the network. Then, the redundancy among these features is reduced by disentangled learning, in which these features are recombined to reconstruct the PAN and LRMS images. Besides, these features should complement each other. So, we utilize the maximal coding rate reduction to balance the redundancy and complementarity among them. Finally, all features are recombined to synthesize the high spatial resolution multispectral image. The experimental results demonstrate that the proposed TDNet has a superior performance in terms of qualitative and quantitative evaluations. The code link is https://github.com/RSMagneto/TDNet.

Minimal and maximal cyclic codes of length 2p

In this paper, we compute the maximal and minimal codes of length 2p over finite fields Fq with p and q are distinct odd primes and f(p) = p – 1 is the multiplicative order of q modulo 2p. We show that, every cyclic code is a direct sum of minimal cyclic codes.

Multi-Device Task-Oriented Communication via Maximal Coding Rate Reduction

Multi-Device Task-Oriented Communication via Maximal Coding Rate Reduction

Optimal Reference for DNA Synthesis

In recent years, DNA has emerged as a potentially viable storage technology. DNA synthesis, which refers to the task of writing the data into DNA, is perhaps the most costly part of existing storage systems. Consequently, the high cost and low throughput limit the practical use of available DNA synthesis technologies. It has been found that the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">homopolymer run</i> (i.e., the repetition of the same nucleotide) is a major factor affecting the synthesis and sequencing errors. Recently, [26] raised and studied the coding problem for efficient synthesis for DNA-based storage systems. Among other things, they studied the maximal code size under synthesis constraints. In [29], the authors studied the role of batch optimization in reducing the cost of large-scale DNA synthesis, for a given pool <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> of random quaternary strings of fixed length. This problem is related to the problem posed in [26] which can be viewed as the opposite side of the coin. Instead of seeking the largest code in which every codeword can be synthesized in a certain amount of time, they asked what is the average synthesis time of a randomly chosen string. Following the lead of [29], in this paper, we take a step forward towards the theoretical understanding of DNA synthesis, and study the homopolymer run of length <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> ≥ 1. Specifically, we are given a set of DNA strands <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> , randomly drawn from a Markovian distribution modeling a general homopolymer run length constraint, that we wish to synthesize. For this problem, we derive asymptotically tight high probability lower and upper bounds on the cost of DNA synthesis, for any <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> ≥ 1. Our bounds imply that, perhaps surprisingly, the periodic sequence ACGT is asymptotically optimal in the sense of achieving the smallest possible cost. Our main technical contribution is the representation of the DNA synthesis process as a certain constrained system, for which string techniques can be applied.

List decoding of maximal order codes over number fields

In this paper, we provide two distinct list decoding algorithms tailored for specific classes of maximal order number field codes. The first algorithm focuses on the unit codes, while the second algorithm is dedicated to the Arakelov divisor-based number field codes. These codes were originally introduced by Maire and Oggier. To achieve our decoding goals, we utilize the list decoding algorithm for general number field codes proposed by Biasse and Quintin. Our approach involves carefully calculating the required parameter B from Biasse and Quintin's algorithm, and adjusting the returned list to satisfy the desired conditions. For unit codes, we consider two cases: when the underlying number field is totally real and when it has an arbitrary signature. We separate these cases because the parameter B can be improved in the totally real scenario. Additionally, we present our list decoding algorithm for Arakelov divisor-based number field codes, focusing on number fields with arbitrary signatures.

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.

Approximate NFA universality and related problems motivated by information theory

In coding and information theory, it is desirable to construct maximal codes that can be either variable length codes or error control codes of fixed length. However deciding code maximality boils down to deciding whether a given NFA is universal, and this is a hard problem (including the case of whether the NFA accepts all words of a fixed length). On the other hand, it is acceptable to know whether a code is ‘approximately’ maximal, which then boils down to whether a given NFA is ‘approximately’ universal. Here we introduce the notion of a (1−ε)-universal automaton and present polynomial randomized approximation algorithms to test NFA universality and related hard automata problems, for certain natural probability distributions on the set of words. We also conclude that the randomization aspect is necessary, as approximate universality remains hard for any fixed polynomially computable ε.

Maximal (v, k, 2, 1) Optical Orthogonal Codes with k = 6 and 7 and Small Lengths

Optical orthogonal codes (OOCs) are used in optical code division multiple access systems to allow a large number of users to communicate simultaneously with a low error probability. The number of simultaneous users is at most as big as the number of codewords of such a code. We consider (v,k,2,1)-OOCs, namely OOCs with length v, weight k, auto-correlation 2, and cross-correlation 1. An upper bound B0(v,k,2,1) on the maximal number of codewords of such an OOC was derived in 1995. The number of codes that meet this bound, however, is very small. For k≤5, the (v,k,2,1)-OOCs have already been thoroughly studied by many authors, and new upper bounds were derived for (v,4,2,1) in 2011, and for (v,5,2,1) in 2012. In the present paper, we determine constructively the maximal size of (v,6,2,1)- and (v,7,2,1)-OOCs for v≤165 and v≤153, respectively. Using the types of the possible codewords, we calculate an upper bound B1(v,k,2,1)≤B0(v,k,2,1) on the code size of (v,6,2,1)- and (v,7,2,1)-OOCs for each length v≤720 and v≤340, respectively.

On Deep Holes of Projective Reed-Solomon Codes over Finite Fields with Even Characteristic

Projective Reed-Solomon code is an important class of maximal distance separable codes in reliable communication and deep holes play important roles in its decoding. In this paper, we obtain two classes of deep holes of projective Reed-Solomon codes over finite fields with even characteristic. That is, let [see formula in PDF] be finite field with even characteristic, [see formula in PDF], and let [see formula in PDF] be the Lagrange interpolation polynomial of the first [see formula in PDF] components of the received vector [see formula in PDF]. Suppose that the [see formula in PDF]-th component of [see formula in PDF] is 0, and [see formula in PDF],[see formula in PDF] where [see formula in PDF], and [see formula in PDF] is a polynomial over [see formula in PDF] with degree no more than [see formula in PDF]. Then the received vector [see formula in PDF] is a deep hole of projective Reed-Solomon codes [see formula in PDF]. In fact, our result partially solved an open problem on deep holes of projective Reed-Solomon codes proposed by Wan in 2020.

НЕРОЗДІЛЬНІ БЛОЧНІ 9-РОЗРЯДНІ ЗАВАДОСТІЙКІ КОДИ ДЛЯ ВИПРАВЛЕННЯ ОДНОРАЗОВОЇ ПОМИЛКИ

With the rapid development of digital telecommunication technologies, the usage of new methods to increase the speed and reliability of data transfer is becoming more relevant. Examples may include data encoding based on the artificial introduction of redundancy and enabling the receiving side not only to detect data distortions but also to form correct values. The separable codes (for example, the Hamming codes) are inferior to non-separable error correction codes in speed. But the increase in the speed of data transmission requires utilising complex, multipoaitional algorithms for searching maximal codes. This leads to an increase in the time of static proof codes determination compared to separable codes, although the speed of data transmission in operation is increased as well, having identical capacities for fault resistant. &nbsp;Technology is proposed that is based on using 9-bit non-separable codes and the encoders/decoders for them, factoring in the fact of absence of the generalised theory of building them. Factoring in the aspects of specialised devices implementation based on an integral technology, the usage of FPGA technology is the most opportune. &nbsp;For practical implementation of universal encodecs/decoders for 9-bit non-separable codes, an approach is proposed utilising FPGA that can be configured to any of these codes with any decimal numbers (the non-separable codes do not contain informational and test parts). Such properties of non-separable codes enable increasing the transmission speed of BCD words. Additionally, the obtained results could be a basis for the further development of a theory and practice for employing the block inseparable codes when the block size increases and it is a reassuring factor as for the analysis of non-separable codes in case of two or more errors correction.