arXiv subjects

Tuvi Etzion

Publications and source records attributed to Tuvi Etzion.

At least 19 recordsLinked to original sources

arXiv2026

Covering Sequences and Covering-Sequences Codes

An $(n,R)_q$-covering sequence is a cyclic sequence, over the finite field $\F_q$, whose consecutive $n$-tuples form a code of length $n$ and covering radius $R$. An $(n,m,R)_q$-covering-sequences code is a set of cyclic sequences of length $m$, over $\F_q$, whose consecutive $n$-tuples form a code of length $n$ and covering radius $R$. These codes are the best building blocks for $(n,R)_q$-covering sequences. We show, for small radii, how cyclic codes and constacyclic codes with small covering radius, can be used to construct such sequences of short length and such codes with a relatively small number of sequences and a total number of codewords in the associated covering code. Sequences with small radius whose length approaches asymptotically to optimality are constructed, especially for an alphabet of prime power size large enough. With the same construction, interesting codes are also constructed for larger radii.

cs.IT↗

arXiv2026

Optimal Non-Binary Single-Track Gray Code

A single-track Gray code is a cyclic Gray code with codewords of length $n$, over an alphabet of size $m$, such that all the $n$ tracks that correspond to the $n$ distinct coordinates of the codewords are cyclic shifts of the first track. Such codes have advantages over the conventional Gray codes in certain quantization and coding applications. Unless $n=2$, there are no such binary codes that contain all the $2^n$ codewords of length $n$. In this paper, we prove that such codes of length $p^t$, $n \geq 2$, with $p^{p^t}$ codewords, over $\F_p$, $p$ prime, exist, for $p=3$ and $p=5$. For larger prime $p$ an appropriate code for $t=2$, implies the existence of such a code for any $t >2$. If the alphabet size $m$ is not a prime there are also indications that such codes exist.

cs.IT↗

arXiv2026

Binary and Non-Binary Self-Dual Sequences and Maximum Period Single-Track Gray Codes

Binary self-dual sequences have been considered and analyzed throughout the years, and they have been used for various applications. Motivated by a construction for single-track Gray codes, we examine the structure and recursive constructions for binary and non-binary self-dual sequences. The feedback shift registers that generate such sequences are discussed. The connections between these sequences and maximum period single-track codes are also discussed. Maximum period non-binary single-track Gray codes of length $p^t$ and period $p^{p^t}$ are constructed. These are the first infinite families of maximum period codes presented in the literature.

cs.IT↗

arXiv2026

Closed Expressions for the Weight Distributions of Codes Associated with Perfect Codes

Perfect codes are arguably the most fascinating structures in combinatorial coding theory, and their classification and weight distribution are of considerable interest. This classification also involves the analysis of some related structures. This paper considers five closely related structures, but all of them have never been tied together before. These structures are 1-perfect codes, extended 1-perfect codes, nearly perfect 1-covering codes, extended nearly perfect 1-covering codes, and one family of completely regular codes (to be called diamond codes). The current work concentrates on the weight distributions of these five families of codes. In the past, some of these weight distributions were not computed, some required heavy tools, and for some only the weight enumerator was presented. We provide complete weight distributions for all five families using some methods that do not require any heavy tools.

math.CO↗

arXiv2026

One-factorizations of complete multipartite graphs with distance constraints

The present paper considers multipartite graphs from the perspective of design theory and coding theory. A one-factor $F$ of the complete multipartite graph $K_{n\times g}$ (with $n$ parts of size $g$) gives rise to a $(g+1)$-ary code ${\cal C}$ of length $n$ and constant weight two. Furthermore, if the one-factor $F$ meets a certain constraint, then ${\cal C}$ becomes an optimal code with minimum distance three. We initiate the study of one-factorizations of complete multipartite graphs subject to distance constraints. The problem of decomposing $K_{n\times g}$ into the largest subgraphs with minimum distance three is investigated. It is proved that, for $n\le g$, the complete multipartite graph $K_{n\times g}$ can be decomposed into $g^2$ copies of the largest subgraphs with minimum distance three. For even $gn$ with $n>g$, it is proved that the complete multipartite graph $K_{n\times g}$ can be decomposed into $g(n-1)$ one-factors with minimum distance three, leaving a small gap of $n$ (in terms of $g$) to be resolved (If $gn$ is odd when $n>g$, no such decomposition of $K_{n\times g}$ exists).

math.CO↗

arXiv2025

New Nonuniform Group Divisible Designs and Mixed Steiner Systems

This paper considers two closely related concepts, mixed Steiner system and nonuniform group divisible design (GDD). The distinction between the two concepts is the minimum Hamming distance, which is required for mixed Steiner systems but not required for nonuniform group divisible $t$-designs. In other words, it means that every mixed Steiner system is a nonuniform GDD, but the converse is not true. A new construction for mixed Steiner systems based on orthogonal arrays and resolvable Steiner systems is presented. Some of the new mixed Steiner systems (also GDDs) depend on the existence of Mersenne primes or Fermat primes. New parameters of nonuniform GDDs derived from large sets of H-designs (which are generalizations of GDDs) are presented, and in particular, many nonuniform group divisible $t$-designs with $t > 3$ are introduced (for which only one family was known before). Some GDDs are with $t > 4$, parameters for which no such design was known before.

math.CO↗

arXiv2025

Mixed Steiner Triples Systems with Shortest Length

We prove that a 3-GDD of type $1^n k^1 \ell^1$, where $n= k \cdot \ell$, with minimum distance 3 exists for every $k$ and $\ell$ such that $n = k \ell$, $k = 1$ or $3~(mod ~ 6)$, and $\ell = 1$ or $3~(mod ~ 6)$. These designs are of the shortest possible length (smallest number of elements) for given $k$ and $\ell$. Other constructions for such triple systems are also presented.

math.CO↗

arXiv2025

On de Bruijn Array Codes Part II: Linear Codes

An M-sequence generated by a primitive polynomial has many interesting and desirable properties. A pseudo-random array is the two-dimensional generalization of an M-sequence. There are non-primitive polynomials all of whose non-zero sequences have the same period. These polynomials generate \emph{sets} of sequences with properties similar to M-sequences. In this paper, a two-dimensional generalization for such sequences is given. This generalization is for a pseudo-random array code, which is a set of $r_1 \times r_2$ arrays in which each $n_1 \times n_2$ nonzero matrix is contained exactly once as a window in one of the arrays. Moreover, these arrays have the shift-and-add property, i.e., the bitwise addition of two arrays (or a nontrivial shift of such arrays) is another array (or a shift of another array) from the code. All the known arrays can be formed by folding sequences generated from an irreducible polynomial or a reducible polynomial whose factors have the same degree and the same exponent. Two proof techniques are used to prove the constructions are indeed of pseudo-random array codes. The first technique is based on another method, different from folding, for constructing some of these arrays. The second technique is a generalization of a known proof technique. This generalization enables the construction of pseudo-random arrays with parameters not known before, and also provides a variety of pseudo-random array codes which cannot be generated by the first method. The two techniques also suggest two different hierarchies between pseudo-random array codes. Finally, two methods to verify whether a folding of sequences, generated by these polynomials, yields a pseudo-random array or a pseudo-random array code, will be presented.

cs.IT↗

arXiv2025

Constructions of Covering Sequences and Arrays

An $(n,R)$-covering sequence is a cyclic sequence whose consecutive $n$-tuples form a code of length $n$ and covering radius $R$. Using several construction methods improvements of the upper bounds on the length of such sequences for $n \leq 20$ and $1 \leq R \leq 3$, are obtained. The definition is generalized in two directions. An $(n,m,R)$-covering sequence code is a set of cyclic sequences of length $m$ whose consecutive $n$-tuples form a code of length~$n$ and covering radius $R$. The definition is also generalized to arrays in which the $m \times n$ sub-matrices form a covering code with covering radius $R$. We prove that asymptotically there are covering sequences that attain the sphere-covering bound up to a constant factor.

math.CO↗

arXiv2025

Steiner Systems over Mixed Alphabet and Related Designs

A mixed Steiner system MS$(t,k,Q)$ is a set (code) $C$ of words of weight $k$ over an alphabet $Q$, where not all coordinates of a word have the same alphabet size, each word of weight $t$, over $Q$, has distance $k-t$ from exactly one codeword of $C$, and the minimum distance of the code $2(k-t)+1$. Mixed Steiner systems are constructed from perfect mixed codes, resolvable designs, large set, orthogonal arrays, and a new type of pairs-triples design. Necessary conditions for the existence of mixed Steiner systems are presented and it is proved that there are no large sets of these Steiner systems.

math.CO↗

arXiv2025

Maximum Size $t$-Intersecting Families and Anticodes

The maximum size of $t$-intersecting families is one of the most celebrated topics in combinatorics, and its size is known as the Erdős-Ko-Rado theorem. Such intersecting families, also known as constant-weight anticodes in coding theory, were considered in a generalization of the well-known sphere-packing bound. In this work we consider the maximum size of $t$-intersecting families and their associated maximum size constant-weight anticodes over alphabet of size $q >2$. It is proved that the structure of the maximum size constant-weight anticodes with the same length, weight, and diameter, depends on the alphabet size. This structure implies some hierarchy of constant-weight anticodes.

math.CO↗

arXiv2024

On de Bruijn Arrays Codes, Part I: Nonlinear Codes

A de Bruijn array code is a set of $r \times s$ binary doubly-periodic arrays such that each binary $n \times m$ matrix is contained exactly once as a window in one of the arrays. Such a set of arrays can be viewed as a two-dimensional generalization of a perfect factor in the de Bruijn graph. Necessary conditions for the existence of such codes are given. Several direct constructions and recursive constructions for such arrays are given. A framework for a theory of two-dimensional feedback shift registers which is akin to (one-dimensional) feedback shift registers is suggested in the process.

cs.IT↗

arXiv2024

Maximum Length RLL Sequences in de Bruijn Graph

Free-space quantum key distribution requires to synchronize the transmitted and received signals. A timing and synchronization system for this purpose based on a de Bruijn sequence has been proposed and studied recently for a channel associated with quantum communication that requires reliable synchronization. To avoid a long period of no-pulse in such a system on-off pulses are used to simulate a \emph{zero} and on-on pulses are used to simulate a \emph{one}. However, these sequences have high redundancy and low rate. To reduce the redundancy and increase the rate, run-length limited sequences in the de Bruijn graph are proposed for the same purpose. The maximum length of such sequences in the de Bruijn graph is studied and an efficient algorithm to construct a large set of these sequences is presented. Based on known algorithms and enumeration methods, maximum length sequence for which the position of each window can be computed efficiently is presented and an enumeration on the number of such sequences is given.

cs.IT↗

arXiv2024

Pairs in Nested Steiner Quadruple Systems

Motivated by a repair problem for fractional repetition codes in distributed storage, each block of any Steiner quadruple system (SQS) of order $v$ is partitioned into two pairs. Each pair in such a partition is called a nested design pair and its multiplicity is the number of times it is a pair in this partition. Such a partition of each block is considered as a new block design called a nested Steiner quadruple system. Several related questions on this type of design are considered in this paper: What is the maximum multiplicity of the nested design pair with minimum multiplicity? What is the minimum multiplicity of the nested design pair with maximum multiplicity? Are there nested quadruple systems in which all the nested design pairs have the same multiplicity? Of special interest are nested quadruple systems in which all the $\binom{v}{2}$ pairs are nested design pairs with the same multiplicity. Several constructions of nested quadruple systems are considered and in particular classic constructions of SQS are examined.

math.CO↗

arXiv2024

On Nearly Perfect Covering Codes

Nearly perfect packing codes are those codes that meet the Johnson upper bound on the size of error-correcting codes. This bound is an improvement to the sphere-packing bound. A related bound for covering codes is known as the van Wee bound. Codes that meet this bound will be called nearly perfect covering codes. In this paper, such codes with covering radius one will be considered. It will be proved that these codes can be partitioned into three families depending on the smallest distance between neighboring codewords. Some of the codes contained in these families will be completely characterized. Other properties of these codes will be considered too. Construction for codes for each such family will be presented, the weight distribution and the distance distribution of codes from these families are characterized. Finally, extended nearly perfect covering code will be considered and unexpected equivalence classes of codes of the three types will be defined based on the extended codes.

cs.IT↗

arXiv2024

Representing Information on DNA using Patterns Induced by Enzymatic Labeling

Enzymatic DNA labeling is a powerful tool with applications in biochemistry, molecular biology, biotechnology, medical science, and genomic research. This paper contributes to the evolving field of DNA-based data storage by presenting a formal framework for modeling DNA labeling in strings, specifically tailored for data storage purposes. Our approach involves a known DNA molecule as a template for labeling, employing patterns induced by a set of designed labels to represent information. One hypothetical implementation can use CRISPR-Cas9 and gRNA reagents for labeling. Various aspects of the general labeling channel, including fixed-length labels, are explored, and upper bounds on the maximal size of the corresponding codes are given. The study includes the development of an efficient encoder-decoder pair that is proven optimal in terms of maximum code size under specific conditions.

cs.IT↗

arXiv2024

Pseduo-Random and de Bruijn Array Codes

Pseudo-random arrays and perfect maps are the two-dimensional analogs of M-sequences and de Bruijn sequences, respectively. We modify the definitions to be applied to codes. These codes are also the two-dimensional analogs of certain factors in the de Bruijn graph. These factors are called zero factors and perfect factors in the de Bruijn graph. We apply a folding technique to construct pseudo-random array codes and examine the minimum distance of the constructed codes. The folding is applied on sequences generated from irreducible polynomials or a product of irreducible polynomials with the same degree and the same exponent. Direct and recursive constructions for de Bruijn array codes are presented and discussed.

cs.IT↗

arXiv2024

Repairing Reed-Solomon Codes with Side Information

We generalize the problem of recovering a lost/erased symbol in a Reed-Solomon code to the scenario in which some side information about the lost symbol is known. The side information is represented as a set $S$ of linearly independent combinations of the sub-symbols of the lost symbol. When $S = \varnothing$, this reduces to the standard problem of repairing a single codeword symbol. When $S$ is a set of sub-symbols of the erased one, this becomes the repair problem with partially lost/erased symbol. We first establish that the minimum repair bandwidth depends on $|S|$ and not the content of $S$ and construct a lower bound on the repair bandwidth of a linear repair scheme with side information $S$. We then consider the well-known subspace-polynomial repair schemes and show that their repair bandwidths can be optimized by choosing the right subspaces. Finally, we demonstrate several parameter regimes where the optimal bandwidths can be achieved for full-length Reed-Solomon codes.

cs.IT↗