A SYNTHESIS METHOD OF BASIC TERNARY BENT-SQUARES BASED ON THE TRIAD SHIFT OPERATOR

Abstract

Practical application of advanced algebraic constructions in modern communication systems based on MC-CDMA (Multi Code Code Division Multiple Access) technology and in cryptography necessitates their further research. One of the most commonly used advanced algebraic construction is the binary bent-function having a uniform amplitude spectrum of the Walsh-Hadamard transform and, accordingly, having the maximal distance from the codewords of affine code. In addition to the binary bent-functions researchers are currently focuses on the development of synthesis methods of their many-valued analogues. In particular, one of the most effective methods for the synthesis of many-valued bent-functions is the method based on the Agievich bent-squares. In this paper, we developed a regular synthesis method of the ternary bent-squares on the basis of an arbitrary spectral vector and the regular operator of the triad shift. The classification of spectral vectors of lengths N = 3 and N = 9 is performed. On the basis of spectral classification more precise definition of many-valued bent-sequences is given, taking into account the existence of the phenomenon of many-valued bent-sequences for the length, determined by odd power of base. The paper results are valuable for practical use: the development of new constant amplitude codes for MC-CDMA technology, cryptographic primitives, data compression algorithms, signal structures, algorithms of block and stream encryption, based on advanced principles of many-valued logic. The developed bent-squares design method is also a basis for further theoretical research: development of methods of the permutation of rows and columns of basic bent-squares and their sign coding, synthesis of composite bent-squares. In addition, the data on the spectral classification of vectors give the task of constructing the synthesis methods of bent-functions of lengths N = 32k+1, k Є ℕ.