Iliya Bouyukliev

• Professor
• Professor (Full) at Bulgarian Academy of Sciences

A large number of algorithms solving Coding Theory problems involve operations on vectors over finite fields. The use of extended CPU registers and instructions is suitable for the optimization of these algorithms. Current work presents the Neon instruction set for the ARM architectures used in Apple's M series of processors. A method for their app...

The covering radius is an important parameter of linear codes. Its calculation is an NP-complete problem. There are three main approaches to its computing that can be considered. The first method is based on traversing the cosets of the linear code. The second method uses the parity-check matrix of the code to calculate the covering radius. The thi...

This work studies projective self-dual (PSD) and self-polar linear codes over finite fields with q elements, where q is a power of a prime. The possible parameters for which PSD codes may exist are presented, and many examples are provided. Algorithms for checking whether a q-ary linear code is self-polar are described. Many PSD and self-polar code...

In this paper, three discrete transforms related to vector spaces over finite fields are studied. For our purposes, and according to the properties of the finite fields, the most suitable transforms are as follows: for binary fields, this is the Walsh–Hadamard transform; for odd prime fields, the Vilenkin–Chrestenson transform; and for composite fi...

In this work, we systematize several implementations of the Gray code over an alphabet with m≥2 elements, which we present in C code so that they can be used directly after copying from the text. We consider two variants—reflected and modular (or shifted) m-ary Gray codes. For both variants, we present the ranking and unranking functions, as well a...

An algorithm for equivalence of linear codes over finite fields is presented. Its main advantage is that it can extract exactly one representative from each equivalence class among a large number of linear codes. It can also be used as a test for isomorphism of binary matrices. The algorithm is implemented in the program LCequivalence, which is des...

Binary codes have a special place in coding theory since they are one of the most commonly used in practice. There are classes of codes specific only to the binary case. One such class is self-complementary codes. Self-complementary linear codes are binary codes that, together with any vector, contain its complement as well. This paper is about bin...

In this paper, we present a library with sequential and parallel functions for computing some of the most important cryptographic characteristics of Boolean and vectorial Boolean functions. The library implements algorithms to calculate the nonlinearity, algebraic degree, autocorrelation, differential uniformity and related tables of vectorial Bool...

Bounds for the parameters of codes are very important in coding theory. The Grey–Rankin bound refers to the cardinality of a self-complementary binary code. Codes meeting this bound are associated with families of two-weight codes and other combinatorial structures. We study the relations among six infinite families of binary linear codes with two...

We present a generalization of Walsh-Hadamard transform that is suitable for applications in Coding Theory, especially for computation of the weight distribution and the covering radius of a linear code over a finite field. The transform used in our research, is a modification of Vilenkin-Chrestenson transform. Instead of using all the vectors in t...

The equivalence test is a main part in any classification problem. It helps to prove bounds for the main parameters of the considered combinatorial structures and to study their properties. In this paper, we present algorithms for equivalence of linear codes, based on their relation to multisets of points in a projective geometry.

The Reed-Muller transform is widely used in discrete mathematics and cryptography, in particular for computing the algebraic normal form of Boolean functions. This is a good reason to look for ways to optimize the implementation of the algorithm. Here we present different ways for optimization based on the bitwise representation of the true table v...

Two algorithms for the classification of linear codes over finite fields are presented. One of the algorithms is based on canonical augmentation and the other one on lattice point enumeration. New classification results over fields with 2, 3 and 4 elements are obtained.

A modification of the Brouwer–Zimmermann algorithm for calculating the minimum weight of a linear code over a finite field is presented. The aim was to reduce the number of codewords for consideration. The reduction is significant in cases where the length of a code is not divisible by its dimensions. The proposed algorithm can also be used to find...

We present methods of using Advanced Vector Extensions (AVX, AVX2, AVX512) for calculation of weight characteristics of binary linear codes. Some experimental results for different lengths and dimensions are presented.

Some aspects of the algorithms for calculating the minimum distance of linear codes over finite fields are presented.

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We classify all binary linear [30, 15, 8] codes with dual distance 8 using the software package QExtNewEdition. There are exactly 42 such codes and all of them are formally self-dual even codes. This result closes the long standing open problem about the existence of a formally self-dual odd code with these parameters.

This paper is devoted to the program Generation which is a self-containing console application for classification of linear codes. It can be used for codes over fields with \(q<8\) elements and with wide-range parameters. The base of the implemented algorithm is the concept of canonical augmentation.

An approach for classification of linear codes with given parameters starting from their proper residual codes or subcodes is presented. The base of the algorithm is the concept of canonical augmentation which is important for parallel implementations. The algorithms are implemented in the programs LengthExtension and DimExtension of the package Qe...

Using an exhaustive computer search, we prove that the number of inequivalent ( 29 , 5 ) -arcs in PG ( 2 , 7 ) is exactly 22. This generalizes a result of Barlotti (see Barlotti, A. Some Topics in Finite Geometrical Structures, 1965), who constructed the first such arc from a conic. Our classification result is based on the fact that arcs and linea...

We present algorithms for classification of linear codes over finite fields, based on canonical augmentation and on lattice point enumeration. We apply these algorithms to obtain classification results over fields with 2, 3 and 4 elements. We validate a correct implementation of the algorithms with known classification results from the literature,...

We consider the problem of computing the weight distribution of a linear code of dimension k over a composite finite field F_q where q = 2^m. Due to the trace function we reduce the arithmetic operations over the composite field to those over a prime field. This allows us to apply a transform of Walsh-Hadamard type. The codes are represented by the...

The aim of this paper is to construct S-boxes of different sizes with good cryptographic properties. An algebraic construction for bijective S-boxes is described. It uses quasi-cyclic representations of the binary simplex code. Good S-boxes of sizes 4, 6, 8, 9, 10, 11, 12, 14, 15, 16 and 18 are obtained.

We propose an algorithm for classification of linear codes over different finite fields based on canonical augmentation. We apply this algorithm to obtain classification results over fields with 2, 3 and 4 elements.

An algorithm for computing the weight distribution of a linear [n, k] code over a finite field F q is developed. The codes are represented by their characteristic vector with respect to a given generator matrix and a generator matrix of the k-dimensional simplex code S q,k .

In this work, we present main combinatorial algorithms implemented in the new version of the software package for linear codes Q-EXTENSION.

We develop an algorithm for computing the weight distribution of a linear $[n,k]$ code over a finite field $\mathbb{F}_q$. We represent the codes by their characteristic vector with respect to a given generator matrix and a special type of a generator matrix of the $k$-dimensional simplex code. This characteristic vector is the input data of our al...

Some of the most important cryptographic characteristics of the Boolean and vector Boolean functions (nonlinearity, autocorrelation, differential uniformity) are connected with the Walsh spectrum. In this paper, we present several algorithms for computing the Walsh spectrum implemented in CUDA for parallel execution on GPU. They are based on the mo...

The classification of combinatorial objects consists of two sub-problems - construction of objects with given properties and rejection of isomorphic objects. In this paper, we consider generation of combinatorial objects that are uniquely defined by a matrix. The method that we present is implemented by backtrack search. The used approach is close...

In this article, we study two representations of a Boolean function which are very important in the context of cryptography. We describeMöbius and Walsh Transforms for Boolean functions in details and presenteffective algorithms for their implementation. We combine these algorithmswith the Gray code to compute the linearity, nonlinearity and algebr...

This paper presents developed software in the area of CodingTheory. Using it, all binary self-dual codes with given properties can beclassified. The programs have consequent and parallel implementations.ACM Computing Classification System (1998): G.4, E.4.

One of the most important cryptographic characteristics of the Boolean and vector Boolean functions is the algebraic degree which is connected with the Algebraic Normal Form. In this paper, we present an algorithm for computing the Algebraic Normal Form of a Boolean function using binary Fast Möbius (Reed-Muller) Transform implemented in CUDA for p...

In this paper we present a new look at the projective dual transform. We consider linear codes represented by their characteristic vector.

In this paper we present a construction for S-boxes using quasi-cyclic codes. We obtain S-boxes with good nonlinearity.

Methods for representing equivalence problems of various combinatorial objectsas graphs or binary matrices are considered. Such representations can be used for isomorphism testing in classification or generation algorithms. Often it is easier to consider a graph or a binary matrix isomorphism problem than to implement heavy algorithms depending esp...

An efficient isomorph-free generation algorithm for classification of binary self-dual codes with the minimum distance $d = 4$ is presented. It is combined with another isomorph-free generation algorithm for classification of self-dual codes with the minimum distance $d ge 6$ . A complete classification of all binary self-dual codes of length 40 is...

We construct a lot of new codes close to the Griesmer bound and prove the nonexistence of some Griesmer codes to determine the exact value of or to improve the known upper bound on , where is the minimum length for which an code exists. We also give the updated table for for all except some known cases.

Purpose ‐ The aim of this paper is to propose the bibliographic information system "Personal eLibrary bgMath". This is a system for the organisation, storage and usage of digitalised literature. It supports the user in organising his/her publications and his/her library of electronic issues. The system is very convenient, especially for scientific...

Self-orthogonal codes are constructed from matrices generated according to parameters of combinatorial designs. An approach towards generating designs and such matrices is considered. Some classification results on self-orthogonal codes are also presented.

In this paper, binary extremal singly even self-dual codes of length 40 and extremal odd unimodular lattices in dimension 40 are studied. We give a classification of extremal singly even self-dual codes of length 40. We also give a classification of extremal odd unimodular lattices in dimension 40 with shadows having 80 vectors of norm 2 through th...

All linear binary codes with minimum distance 8 and codimension up to 14 and all codes with minimum distance 10 and codimension up to 18 are classified. Nonexistence of codes with parameters [33,18,8] and [33,14,10] is proved. This leads to 8 new exact bounds for binary linear codes. Primarily two algorithms considering the dual codes are used, nam...

An efficient algorithm for classification of binary self-dual codes is presented. As an application, a complete classification of the self-dual codes of length 38 is given.

It is known that there are extremal formally self-dual even codes which are not self-dual only for lengths 6, 10, 12, 14, 18, 20, 22, 28 and 30. We complete the classification of extremal formally self-dual even codes by examining the case for length 30.

This paper considers probabilistic algorithms for minimum distance of linear codes. Two implementations of a probabilistic algorithm of Leon's type are developed. Some experimental results for our implementations are presented.

In this work, we consider a classification of infinite families of linear codes which achieve the Griesmer bound, using the projective dual transform. We investigate the correspondence between families of linear codes with given properties via dual transform.

We present the full classification of Hadamard 2-(31,15,7), Hadamard 2-(35, 17,8) and Menon 2-(36,15,6) designs with automorphisms of odd prime order. We also give partial classifications of such designs with automorphisms of order 2. These classifications lead to related Hadamard matrices and self-dual codes. We found 76166 Hadamard matrices of or...

The aim of this work is a systematic investigation of the possible parameters of quasi-perfect (QP) binary and ternary linear codes of small dimensions and preparing a complete classification of all such codes. First, we give a list of infinite families of QP codes which includes all binary, ternary, and quaternary codes known to us. We continue fu...

The problem of computing the number of codewords of weights not exceeding a given integer in linear codes over a finite field is considered. An efficient method for solving this problem is proposed and discussed in detail. It builds and uses a sequence of different generator matrices, as many as possible, so that the identity matrix takes disjoint...

The problem of ecien t computing of the ane vector oper- ations (addition of two vectors and multiplication of a vector by a scalar over GF (q)), and also the weight of a given vector, is important for many problems in coding theory, cryptography, VLSI technology etc. In this paper we propose a new way of representing vectors over GF (3) and GF (4)...

In this work a heuristic algorithm for obtaining lower bounds on the covering radius of a linear code is developed. Using this algorithm the least covering radii of the binary linear codes of dimension 6 are determined. Upper bounds for the least covering radii of binary linear codes of dimensions 8 and 9 are derived.

In this paper we present a developed software in the area of Coding Theory. Using it, codes with given properties can be classied. A part of this software can be used also for investigations (isomorphisms, automorphism groups) of other discrete structures|combinatorial designs, Hadamard matrices, bipartite graphs etc.

The number of known inequivalent binary self-complementary [120, 9, 56] codes (and hence the number of known binary self-complementary [136, 9, 64] codes) is increased from 25 to 4668 by showing that there are exactly 4650 such inequivalent codes with an automorphism of order 3. This implies that there are at least 4668 nonisomorphic quasi-symmetri...

In this paper we discuss an algorithm for code equivalence. We reduce the equivalence test for linear codes to a test for isomorphism of binary matrices.

All binary projective codes of dimension up to 6 are classified. Information about the number of the codes with different minimum distances and automorphism group orders is given.

In this paper, we complete the classification of optimal binary linear self-orthogonal codes up to length 25. Optimal self-orthogonal codes are also classified for parameters up to length 40 and dimension 10. The results were obtained via two independent computer searches.

We prove that a linear binary code with parameters [34,24,5] does not exist. Also, we characterize some codes with minimum distance 5 and 6.

Projective two-weight codes with relatively small parameters are enumerated up to equivalence. Some properties of codes and related strongly-regular graphs are presented.

International audience Before this work, at least 762 inequivalent Hadamard matrices of order 36 were known. We found 7238 Hadamard matrices of order 36 and 522 inequivalent [72,36,12] double-even self-dual codes which are obtained from all 2-(35,17,8) designs with an automorphism of order 3 and 2 fixed points and blocks.

Several methods for classifying self-orthogonal codes up to equiva- lence are presented. These methods are used to classify self-orthogonal codes with largest possible minimum distance over the fields F3 and F4 for lengths n 29 and small dimensions (up to 6). Some properties of the classified codes are also presented. In particular, an extensive co...

Because of their practical significance linear block codes are traditionally included in basic Coding Theory courses. In this paper, using a rich set of simple examples, we demonstrate some interesting facts which, we believe, are useful to get to know for students, and will help them to learn how to solve practical error control problems in modern...

Let n4(k,d) be the minimum length of a linear [n,k,d] code over GF(4) for given values of k and d. For codes of dimension five, we compute the exact values of n4(5,d) for 75 previously open cases. Additionally, we show that n4(6,14)=24, n4(7,9)=18, and n4(7,10)=20. Moreover, we classify optimal quaternary codes for some values of n and k.

All ternary projective codes of dimension 4 and these of dimension 5 of lengths up to 15 are classified. Their automorphism groups and weight spectra are determined. The lest value of the covering radius of ternary codes of dimension 4 and lengths between 13 and 20 are computed. I Preliminaries In this work we investigate ternary projective codes o...

We present some results on almost maximum distance separable (AMDS) codes and Griesmer codes of dimension 4 over over the field of order 5. We prove that no AMDS code of length 13 and minimum distance 5 exists, and we give a classification of some AMDS codes. Moreover, we classify the projective strongly optimal Griesmer codes over F5 of dimension...

Let d₃(n,k) be the maximum possible minimum Hamming distance of a ternary [n,k,d]-code for given values of n and k. We describe a package for code extension and use this to prove some new exact values of d₃(n,k). Moreover, we classify the ternary [n,k,d₃(n,k)]-codes for some values of n and k

We classify optimal [n,k,d] binary linear codes of dimension ⩽7, with one exception, where by optimal we mean that no [n−1,k,d],[n+1,k+1,d], or [n+1,k,d+1] code exists. In particular, we present (new) classification results for codes with parameters [40,7,18], [43,7,20], [59,7,28], [75,7,36], [79,7,38], [82,7,40], [87,7,42], and [90,7,44]. These cl...

Let n(8,d) be the smallest integer n for which a binary linear code of length n, dimension 8, and minimum distance d exists. We prove that n(8,18)=42, n(8,26)=58, n(8,28)=61, n(8,30)=65, n(8,34)=74, n(8,36)=77, n(8,38)=81, n(8,42)=89, and n(8,60)=124. After these results, all values of n(8,d) are known

We prove the nonexistence of binary [69,9,32] codes and construct codes with parameters [76,9,34],[297,9,146], and [300,9,148]. These results show that n(9,32)=70, n(9,34)&les;76,n(9,146)=297, and n(9,148)=300, where n(k,d) denotes the smallest value of n for which there exists an [n,k,d] binary code. We also present some codes of minimum distance...

We construct extremal self-dual codes with lengths 44, 50, 54, and 58. They have weight enumerators for which extremal codes were previously not known to exist. Two methods are used for constructing the codes using self dual codes of same or smaller length. To obtain the codes we use a combinatorial optimization search

A method to design binary self-dual codes with an automor-phism of order two without fixed points is presented. Extremal self-dual codes with lengths 40; 42; 44; 54; 58; 68 are constructed. Many of them have weight enumerators for which extremal codes were previously not known to exist. Index Terms—Automorphism, heuristic algorithm, self-dual codes...

Constructions of [162,8,80] and [159,8,78] codes are given. This solves the open problems of finding the minimum length of binary codes of dimension 8 and minimum distances 78 and 80 respectively. Keywords: Optimal binary codes 1 Introduction Let F n q be the n-dimensional vector space over the Galois field F q . The Hamming distance between two ve...

Let d3(n,k) be the maximum possible minimum Hamming distance of a ternary [ n,k,d;3]-code for given values of n and k. It is proved that d3(44,6)=27, d3(76,6)=48,d3(94,6)=60 , d3(124,6)=81,d3(130,6)=84 , d3(134,6)=87,d3(138,6)=90 , d3(148,6)=96,d3(152,6)=99 , d3(156,6)=102,d3(164,6)=108 , d3(170,6)=111,d3(179,6)=117 , d3(188,6)=123,d3(206,6)=135 ,...

Let n4(k, d) be the smallest integer n, such that a quaternary linear [n, k, d]-code exists. The bounds n4(5, 21) ⩽ 32, n4(5, 30) = 43, n4(5, 32) = 46, n4(5, 36) = 51, n4(5,40) ⩽ 57, n4(5, 48) ⩽ 67, n4(5, 64) = 88, n4(5, 68) ⩽ 94, n4(5, 70) ⩽ 97, n4(5, 92) ⩽ 126, n4(5, 98) ⩽ 135, n4(5, 122) = 165, n4(5, 132) ⩽ 179, n4(5, 136) ⩽ 184, n4(5, 140) = 18...

Let n_q(k,d) be the smallest integer n for which there exists a linear code of length n, dimension k, and minimum distance d, over a field of q elements. In this correspondence we determine n₅(4,d) for all but 22 values of d

Constructions of [162,8,80] and [159,8,78] codes are given. This solves the open problemsof finding the minimum length of binary codes of dimension 8 and minimum distances78 and 80 respectively.Keywords: Optimal binary codes1 IntroductionLet Fnq be the n-dimensional vector space over the Galois field F q . The Hamming distancebetween two vectors of...

Binary linear codes with parameters [45,14,16], [76,9,34] and [94,12,40] have been constructed. Codes with such parameters were not known until now

Let d, (n k ) be the maximum possible minimum Hamming It is proved that d4 (33,5) = 22, d4(49 5 ) = 34, &(I31 5) = 96, d4(142,5) = 104, rla(147,5) = 108, &(I52 5 ) = 112, &(I58 5 ) =116,d4(176,5) 2 129,d4(180,5) 2 132,&(190 5 ) 2 140,&(19j 5) =144,d4(200,5) = 148,d4(205 5) = 132,d4(216 3) = 160,d4(22i 2) == 180, and d4(247,5) = 184. A survey of the...

Linear codes with parameters [47,5,30;3], [44,6,27;3], [90,6,57;3] and [94,6,60;3] have been found

A method to design good linear codes is presented. Some new linear block codes over F 3 and F 4 are found.

In this paper we study the function N(d;d?). More precisely, we give results for classication and construction of codes which reach, or give upper bounds on N(d;d?).

An algorithm for computing the weight distribution of the coset leaders is proposed. This restric- tive search algorithm is the most eective to the computation of the weight distribution of the coset leaders for codes of rate close to 1 2.

We present a method for classification of codes, using known part of their generator matrices. There are two main problems: the finding of all codes with parameters [n, k, d], and their partition into different classes of equivalence. The corresponding algorithms consists of different combinatorial and optimization ideas. With their realization, th...

An efficient algorithm for classification of binary self-dual codes with minimum distance four is presented.