Volume 35, Issue 2

This paper is concerned with images of cosets in the direct product of groups by bijective mappings from factors to groups.We prove necessary and sufficient conditions on bijective mappings for existence a coset in the direct product of two groups whose image is a coset. Under some constraints on bijective mappings, we describe the cosets in the direct product of groups, whose images by bijective mappings from factors to groups are cosets, We described the cosets in the direct product of elementary abelian 2-groups whose images by the inversion permutation of nonzero elements of a finite field on factors are cosets. We also describe the similar cosets for the permutation used an an s-box of Kuznyechik algorithm, Under some constraints on bijective mappings, we describe the automorphisms of the direct product groups which commute with bijective mappings from factors to groups.

Let p be a prime number p ≥3. We consider the set of decompositions of a p -logic function into a sum of functions with disjoint subsets of variables obtained by means of linear substitutions of arguments. Each decomposition of this kind is associated with a decomposition of the vector space into a direct sum of subspaces. We present conditions under which such space decomposition is unique up to rearrangement of subspaces.

Series connected codes are generated by matrices obtained by series connected generating matrices of other linear codes. We find an estimate for the probability that the Hadamard square of series connected random linear codes coincides with the Cartesian product of the Hadamard squares of the linear codes involved in the connection.

The nonlinearity of a vectorial function over a finite field is defined in the paper as the Hamming distance from the function to the set of affine mappings in the space of values of all vectorial functions. For an arbitrary field of q elements we derive lower bounds for the nonlinearity of PN and APN functions in n variables in the form qn−qn−3⋅2−2−2−1$ {q^n} - \sqrt {{q^n} - 3 \cdot {2^{ - 2}}} - {2^{ - 1}} $ and qn−2qn−7⋅2−2−2−1$ {q^n} - \sqrt {2{q^n} - 7 \cdot {2^{ - 2}}} - {2^{ - 1}} $, respectively. These bounds improve the estimates obtained earlier in the Boolean case. It is shown that the nonlinearity of such functions can be estimated from above by q n − n − 1. For q = 2, 3, 4 the exact values of the nonlinearity of PN and APN functions of low dimension are obtained.

The closed class Polpm $ {\rm Pol}_{p^m} $ in p m -valued logic, where p is a prime number, 1 ⩽  m ⩽ p is studied. This class consists of all functions that are polynomial modulo p m . Criteria for polynomiality modulo p m of a function in p m -valued logic are found. A relation describing the class Polpm $ {\rm Pol}_{p^m} $ is obtained in an explicit form.