| Popis: |
A (periodic) correlation function is an important measure to evaluate the effectiveness of pseudorandom sequences. In practice, the sequences are required to have the impulse-like autocorrelation function. Also, crosscorrelation magnitudes of the distinct sequences must be as low as possible. Theoretically, the correlation of sequences has a strong connection with problems of algebraic coding and combinatorics. Namely, the correlation distribution of sequences is related to the weight distribution of codewords in algebraic codes. Furthermore, binary sequences with ideal two-level autocorrelation are equivalent to cyclic Hadamard difference sets in combinatorics. Therefore, a good knowledge of coding theory and combinatorics is helpful for a study of sequences with low correlation. In this thesis, the correlation of binary sequences is studied by the aid of fruitful results of coding theory and combinatorics. From this study, some interesting properties are presented on the correlation of binary sequences, and several new binary sequences with low correlation are discovered. The outline of thesis is as follows. First of all, crosscorrelation properties of binary sequences with ideal two-level autocorrelation are studied. The 3- and 5-valued crosscorrelation properties of several classes of binary sequences with ideal two-level autocorrelation are completely determined by either theoretical proofs or conjectures. Second, new binary sequences of period of a multiple of 4 are proposed by making use of the interleaved structure of sequences. The sequences have three-valued out-of-phase autocorrelation, i.e., {0, -4, +4}, which is optimal with respect to autocorrelation magnitude. Third, a new binary sequence family with low correlation and a large family size is proposed and its asymptotic optimality is examined. With respect to maximum correlation, a family size, and a linear complexity, the sequence family is competitive among all known binary sequence families. Finally, the construction of quadratic bent functions of a special polynomial form is presented by providing necessary and sufficient conditions on the coefficients of the polynomial. The resulting quadratic bent functions can be employed for the linear feedback shift register (LFSR) implementation of a family of bent sequences, which is a potential candidate for future code-division multiple access (CDMA) system. |
