Cardinality Of Codes Research Articles

Bounds on the Maximum Cardinality of Indel and Substitution Correcting Codes

Bounds on the Maximum Cardinality of Indel and Substitution Correcting Codes

Improved bounds for codes correcting insertions and deletions

AbstractThis paper studies the cardinality of codes correcting insertions and deletions. We give improved upper and lower bounds on code size. Our upper bound is obtained by utilizing the asymmetric property of list decoding for insertions and deletions and can be seen as analogous to the Elias bound in the Hamming metric. Our non-asymptotic bound is better than the existing bounds when the minimum Levenshtein distance is relatively large. The asymptotic bound exceeds the Elias and the MRRW bounds adapted from the Hamming-metric bounds for the binary and the quaternary cases. Our lower bound improves on the bound by Levenshtein, but its effect is limited and vanishes asymptotically.

Weight Enumerators and Cardinalities for Number-Theoretic Codes

The number-theoretic codes are a class of codes defined by single or multiple congruences. These codes are mainly used for correcting insertion and deletion errors, and for correcting asymmetric errors. This paper presents a formula for a generalization of the complete weight enumerator for the number-theoretic codes. This formula allows us to derive the weight enumerators and cardinalities for the number-theoretic codes. As a special case, this paper provides the Hamming weight enumerators and cardinalities of the non-binary Tenengolts' codes, correcting single insertion or deletion. Moreover, we show that the formula deduces the MacWilliams identity for the linear codes over the ring of integers modulo $r$.

New linear codes derived from skew generalized quasi-cyclic codes of any length

This article studies skew generalized quasi-cyclic codes (SGQC-codes) over finite fields for any length n. We derive generator polynomials and cardinality of SGQC codes. Moreover, we show that the dual of any length SGQC-code is also an SGQC-code. Our search results lead to the construction of fifteen new 2-generator SGQC codes over the finite field F4 with minimum distances exceeding the minimum distances of the previously best known F4-linear codes with comparable parameters.

Codes for Multiple-Access Asynchronous Techniques

This article discusses asynchronous techniques with spreading sequences (codes) that are statistically uncorrelated for arbitrarily random starting points. We provide improved codes for the spread-spectrum multiple-access asynchronous techniques. The article provides methods for generating improved codes, as well as estimates of their parameters. We investigate the cross-correlation of codes for arbitrarily random starting points. The comparative analysis shows that with a slight increase in the correlation modulus, we can significantly increase the cardinality of the spreading codes. This useful property can be used in future radio control systems and future smart grids to implement soft capacity, i.e. the base station can increase the subscriber capacity with a slight decrease in the quality of service.

Bounds for spherical codes: The Levenshtein framework lifted

Based on the Delsarte-Yudin linear programming approach, we extend Levenshtein's framework to obtain lower bounds for the minimum $h$-energy of spherical codes of prescribed dimension and cardinality, and upper bounds on the maximal cardinality of spherical codes of prescribed dimension and minimum separation. These bounds are universal in the sense that they hold for a large class of potentials $h$ and in the sense of Levenshtein. Moreover, codes attaining the bounds are universally optimal in the sense of Cohn-Kumar. Referring to Levenshtein bounds and the energy bounds of the authors as ``first level", our results can be considered as ``next level" universal bounds as they have the same general nature and imply necessary and sufficient conditions for their local and global optimality. For this purpose, we introduce the notion of Universal Lower Bound space (ULB-space), a space that satisfies certain quadrature and interpolation properties. While there are numerous cases for which our method applies, we will emphasize the model examples of $24$ points ($24$-cell) and $120$ points ($600$-cell) on $\mathbb{S}^3$. In particular, we provide a new proof that the $600$-cell is universally optimal, and in so doing, we derive optimality of the $600$-cell on a class larger than the absolutely monotone potentials considered by Cohn-Kumar.

On two-weight codes

We consider q-ary (linear and nonlinear) block codes with exactly two distances: d and d+δ. We derive necessary conditions for existence of such codes (similar to the known conditions in the projective case). In the linear (but not necessary projective) case, we prove that under certain conditions the existence of such linear 2-weight code with δ>1 implies the following equality of greatest common divisors: (d,q)=(δ,q). Upper bounds for the maximum cardinality of such codes are derived by linear programming and from few-distance spherical codes. Tables of lower and upper bounds for small q=2,3,4 and qn<50 are presented.

Universal Bounds for Size and Energy of Codes of Given Minimum and Maximum Distances

We employ signed measures that are positive definite up to certain degrees to establish Levenshtein-type upper bounds on the cardinality of codes with given minimum and maximum distances, and universal lower bounds on the potential energy (for absolutely monotone interactions) for codes with given maximum distance and cardinality. The distance distributions of codes that attain the bounds are found in terms of the parameters of Levenshtein-type quadrature formulas. Necessary and sufficient conditions for the optimality of our bounds are derived. Further, we obtain upper bounds on the energy of codes of fixed minimum and maximum distances and cardinality.

Perfect Multi Deletion Codes Achieve the Asymptotic Optimality of Code Size

This paper studies on the cardinality of perfect multi deletion binary codes. The lower bound for any perfect deletion code with the fixed code length and the number of deletions, and the asymptotic achievable of Levenshtein's upper bound are shown.

Applications of Gaussian Binomials to Coding Theory for Deletion Error Correction

We present new applications on $q$-binomials, also known as Gaussian binomial coefficients. Our main theorems determine cardinalities of certain error-correcting codes based on Varshamov-Tenengolts codes and prove a curious phenomenon relating to deletion sphere for specific cases.

On q-ary Codes with Two Distances d and d + 1

The $q$-ary block codes with two distances $d$ and $d+1$ are considered. Several constructions of such codes are given, as in the linear case all codes can be obtained by a simple modification of linear equidistant codes. Upper bounds for the maximum cardinality of such codes is derived. Tables of lower and upper bounds for small $q$ and $n$ are presented.

Linear-time algorithms for three domination-based separation problems in block graphs

The problems of determining minimum identifying, locating-dominating or open locating-dominating codes are special search problems that are challenging both from a theoretical and a computational point of view, even for several graph classes where other in general hard problems are easy to solve, like bipartite graphs or chordal graphs. Hence, a typical line of attack for these problems is to determine minimum codes of special graphs. In this work we study the problem of determining the cardinality of minimum such codes in block graphs (that are diamond-free chordal graphs). We present linear-time algorithms for these problems, as a generalization of a linear-time algorithm proposed by Auger in 2010 for identifying codes in trees. Thereby, we provide a subclass of chordal graphs for which all three problems can be solved in linear time.

The Identifying Code, the Locating-dominating, the Open Locating-dominating and the Locating Total-dominating Problems Under Some Graph Operations

The problems of determining minimum identifying, locating-dominating, open locating-dominating or locating total-dominating codes in a graph G are variations of the classical minimum dominating set problem in G and are all known to be hard for general graphs. A typical line of attack is therefore to determine the cardinality of minimum such codes in special graphs. In this work we study the change of minimum such codes under three operations in graphs: adding a universal vertex, taking the generalized corona of a graph, and taking the square of a graph. We apply these operations to paths and cycles which allows us to provide minimum codes in most of the resulting graph classes.

New LMRD Code Bounds for Constant Dimension Codes and Improved Constructions

We prove upper bounds for the cardinality of constant dimension codes (CDC) which contain a lifted maximum rank distance (LMRD) code as a subset. Thereby we cover all parameters fulfilling k <; 3d/2, where k is the codeword dimension and d is the minimum subspace distance. The proofs of the bounds additionally show that an LMRD code L can be unioned with a CDC C (of fitting parameters) without violating the subspace distance condition iff each codeword of C intersects the special subspace of L in at least dimension d/2. This connection is used to find the new largest and sometimes bound achieving CDCs for small parameters.

Energy bounds for codes in polynomial metric spaces

In this article we present a unified treatment for obtaining bounds on the potential energy of codes in the general context of polynomial metric spaces (PM-spaces). The lower bounds we derive via the linear programming techniques of Delsarte and Levenshtein are universally optimal in the sense that they apply to a broad class of energy functionals and, in general, cannot be improved for the specific subspace. Tests are presented for determining whether these universal lower bounds (ULB) can be improved on larger spaces. Our ULBs are applicable on the Euclidean sphere, infinite projective spaces, as well as Hamming and Johnson spaces. Asymptotic results for the ULB for the Euclidean spheres and the binary Hamming space are derived for the case when the cardinality and dimension of the space grow large in a related way. Our results emphasize the common features of the Levenshtein’s universal upper bounds for the cardinality of codes with given separation and our ULBs for energy. We also introduce upper bounds for the energy of designs in PM-spaces and the energy of codes with given separation.

Exact values for three domination-like problems in circular and infinite grid graphs of small height

In this paper we study three domination-like problems, namely identifying codes, locating-dominating codes, and locating-total-dominating codes. We are interested in finding the minimum cardinality of such codes in circular and infinite grid graphs of given height. We provide an alternate proof for already known results, as well as new results. These were obtained by a computer search based on a generic framework, that we developed earlier, for the search of a minimum labeling satisfying a pseudo-d-local property in rotagraphs.

Coding Over Sets for DNA Storage

In this paper we study error-correcting codes for the storage of data in synthetic deoxyribonucleic acid (DNA). We investigate a storage model where a data set is represented by an unordered set of $M$ sequences, each of length $L$ . Errors within that model are a loss of whole sequences and point errors inside the sequences, such as insertions, deletions and substitutions. We derive Gilbert-Varshamov lower bounds and sphere packing upper bounds on achievable cardinalities of error-correcting codes within this storage model. We further propose explicit code constructions than can correct errors in such a storage system that can be encoded and decoded efficiently. Comparing the sizes of these codes to the upper bounds, we show that many of the constructions are close to optimal.

On spherical codes with inner products in a prescribed interval

We develop a framework for obtaining linear programming bounds for spherical codes whose inner products belong to a prescribed subinterval $$[\ell ,s]$$ of $$[-\,1,1)$$ . An intricate relationship between Levenshtein-type upper bounds on cardinality of codes with inner products in $$[\ell ,s]$$ and lower bounds on the potential energy (for absolutely monotone interactions) for codes with inner products in $$[\ell ,1)$$ (when the cardinality of the code is kept fixed) is revealed and explained. Thereby, we obtain a new extension of Levenshtein bounds for such codes. The universality of our bounds is exhibited by a unified derivation and their validity for a wide range of codes and potential functions.

Optimal conflict-avoiding codes for 3, 4 and 5 active users

Conflict-avoiding codes are used in multiple-access collision channels without feedback. The number of codewords in a conflict-avoiding code is the number of potential users of the channel. That is why codes with maximum cardinality (optimal codes) for given parameters are of interest. In this paper we classify, up to multiplier equivalence, all optimal conflict-avoiding codes of weights 3, 4, and 5 and given small lengths. We also determine some previously unknown values of the maximum cardinality of conflict-avoiding codes of weights 4 and 5.

Non-Overlapping Codes

We say that a $q$-ary length $n$ code is \emph{non-overlapping} if the set of non-trivial prefixes of codewords and the set of non-trivial suffices of codewords are disjoint. These codes were first studied by Levenshtein in 1964, motivated by applications in synchronisation. More recently these codes were independently invented (under the name \emph{cross-bifix-free} codes) by Baji\'c and Stojanovi\'c. We provide a simple construction for a class of non-overlapping codes which has optimal cardinality whenever $n$ divides $q$. Moreover, for all parameters $n$ and $q$ we show that a code from this class is close to optimal, in the sense that it has cardinality within a constant factor of an upper bound due to Levenshtein from 1970. Previous constructions have cardinality within a constant factor of the upper bound only when $q$ is fixed. Chee, Kiah, Purkayastha and Wang showed that a $q$-ary length $n$ non-overlapping code contains at most $q^n/(2n-1)$ codewords; this bound is weaker than the Levenshtein bound. Their proof appealed to the application in synchronisation: we provide a direct combinatorial argument to establish the bound of Chee \emph{et al}. We also consider codes of short length, finding the leading term of the maximal cardinality of a non-overlapping code when $n$ is fixed and $q\rightarrow \infty$. The largest cardinality of non-overlapping codes of lengths $3$ or less is determined exactly.