We consider the mappings of a set of words into itself, and describe the injective mappings which do not propagate such errors as alteration of letters in words, the bijective mappings which do not propagate such errors as deletion of letters, and the injective mappings which do not propagate the errors of the above two types.
With the help of the bounded superposition (which allows substitutions of non-selector functions only instead of the first variable), concepts of α-closed class and α-complete system of fc-valued logic functions are defined in the usual way. For k > 7, it is proved that every system which contains all permutations and any quasi-group function is α-complete.
We establish that the Weyl upper bounds for sums of additive multiplicative and mixed characters are attainable in all Galois extensions of the finite fields F q with cyclic Galois group of order v > I, The A. Weyl upper bound for sums of characters over the finite field F q is well known. Naturally, the question on the accuracy of the Weyl estimate arises. This question is reduced to the problem of finding lower bounds for the sums close to the Weyl estimate. Non-trivial lower bounds for rational exponential sums over the prime fields F p were obtained in [1][2][3][4][5]. I-ater Stepanov [6] suggested an elementary proof of the results of [2, 3] and generalized them to incomplete sums.The previously mentioned problem is closely related to one of the main problems of coding theory, the problem of extremal packings and coverings of the Hamming space F g n over F q (see, for example [7-9]). In particular, using results of coding theory, Levenshtein [10] slightly generalized the result of [6]. Using some properties of generalized ReedMuller codes, Bassalygo, Zinovyev and Litsyn proved [11] the attainability of the Weyl upper bound for rational exponential sums with prime denominator for an arbitrary field F q n with odd n > 3 and p > 2. A similar result for some special values of n is obtained in [12].Stepanov in [13] gave a simple and completely arithmetical proof of the result of [11] with an explicit form of the corresponding polynomials and obtained similar results for generalized Legendre symbols.In this paper we extend the results of [13] to the multiplicative characters of arbitrary order and to some mixed sums.
В работе изучаются матрицы переходов биграмм для систем r s G hG подстановок степени 2 n , где t G-группа сдвигов в прямой сумме групп (/ 2 t Z ,+). Частично подтверждена известная гипотеза об отсутствии APNподстановок поля () 2 n GF при четном n , предл ожены способы построения разностно-4-однородных подстановок. Ключевые слова: модулярная группа, разностная характеристика системы подстановок, APN-функция On the matrices of transitions of differences for some modular groups
In the theory of quasigroups it often occurs that the Cayley table of a finite quasigroup can be partitioned into lesser Latin squares. In this paper we introduce a concept of T-partition of a quasigroup which is a generalization of the aforesaid situations. On the set R(Q) of all T-partitions of a quasigroup Q we introduce a relation of partial order and prove that R(Q) is a lattice. In the lattice R(Q) we consider the sublattices corresponding to right-regular, left-regular, regular, homogeneous T-partitions. The lattices of regular and homogeneous T-partitions of a group are isomorphic to the lattices of subgroups and normal subgroups respectively. The lattices of left-regular (right-regular) T-partitions of a group are isomorphic to the lattice of all partitions of the group into left (right) cosets of the system of its subgroups. This lattice contains more information than the lattice of subgroups: any non-simple Abelian group is uniquely determined by its lattice of left-regular (right-regular) T-partitions.
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