Volume 18 (2018): Edition 5 (May 2018)
Special Thematic Edition on Optimal Codes and Related Topics

This volume contains selected papers from the Eighth International Workshop on Optimal Codes and Related Topics (OCRT’2017) organized as minisymposium during the conference Mathematical days in Sofia (10-14 July 2017). This workshop was organized by the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences. There were 37 participants and 26 contributed talks.

We classify tight optimal conflict-avoiding codes of weights 3, 4, 5 and 6 and given small lengths.

In the paper are studied methods for protection of interactive systems and digital archives against unauthorized distribution of digital content. We make an overview of steganographic methods and image protection schemes. An improved watermark error correction scheme is presented. The studies contribute to the overall development of the North+region, provide future generations with widespread public access to digital materials. We achieve a long-term storage, secure data protection and interactive web presence.

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 most popular sequential algorithm. The algorithms differ in the complexity of implementations, resources used, optimization strategies and techniques. In the end, we give some experimental results.

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 to dynamic programming.

We propose the concepts of almost complete subset of an elliptic quadric in the projective space PG(3, q) and of almost complete cap in the space PG(N, q), N ≥ 3, as generalizations of the concepts of almost complete subset of a conic and of almost complete arc in PG(2, q). Upper bounds of the smallest size of the introduced geometrical objects are obtained by probabilistic and algorithmic methods.

Binary codes created by doubling construction, including quasi-perfect ones with distance d = 4, are investigated. All [17·2^r−6, 17·2^r−6 − r, 4] quasi-perfect codes are classified. Weight spectrum of the codes dual to quasi-perfect ones with d = 4 is obtained. The automorphism group Aut(C) of codes obtained by doubling construction is studied. A subgroup of Aut(C) is described and it is proved that the subgroup coincides with Aut(C) if the starting matrix of doubling construction has an odd number of columns. (It happens for all quasi-perfect codes with d = 4 except for Hamming one.) The properness and t-properness for error detection of codes obtained by doubling construction are considered.

The following paper adapts the classical Zigangirov-Jelinek algorithm to the decoding of nonbinary block codes under severe mixed jamming. To ensure reliable communications in this scenario we combine reception techniques based on distribution free statistical tests with sequential decoding on syndrome trellises. It will be shown that the proposed approach can ensure relatively high transmission rate with reasonable complexity.

The (v, u×c, λ)-splitting balanced incomplete block designs correspond to c-splitting authentication codes. We classify splitting balanced incomplete block designs with definite parameters.