Abstract: First, a dual basis B? of B of modulo N is obtained by classical computation. Next, quantum computation is performed using the periodicity of a point sequence included in a sum set of sets obtained by parallel translation of a lattice L(B) by integral multiples of t for a plurality of integers, and an n-dimensional rj=(rj1, . . . , rjn) and rj0 are obtained for j=1, . . . , m. Subsequently, by classical computation, the closest vector rj(c)=(rj1(c), . . . , rjn(c))?L(B?) of the n-dimensional vector rj, and the difference vector rj(d)=rj?rj(c)=(rj1(d), . . . , rjn(d)) corresponding to rj(c) are obtained.

Abstract: First, a dual basis B? of B of modulo N is obtained by classical computation. Next, quantum computation is performed using the periodicity of a point sequence included in a sum set of sets obtained by parallel translation of a lattice L(B) by integral multiples of t for a plurality of integers, and an n-dimensional rj=(rj1, . . . , rjn) and rj0 are obtained for j=1 , . . . , m. Subsequently, by classical computation, the closest vector rj(c)=(rj1(c), . . . , rjn(c))?L(B?) of the n-dimensional vector rj, and the difference vector rj(d)=rj?rj(c)=(rj1(d), . . . , rjn(d)) corresponding to rj(c) are obtained.

Abstract: An information encoding method in which a common multiplier A of the input signal samples x(i) in each frame is estimated. Each sample x(i) is divided by A, the quotient is rounded to the number of significant digits, and the rounded quotient y(i) is encoded. The y(i) is multiplied by A and the product is subtracted from x(i) to obtain an error signal z(i), which is encoded. The A is encoded and the codes of y(i), z(i), and A are combined together and outputted. To estimate A, each numeric value is divided by a determined representative value. Irreducible fractions between the maximum positive error and the maximum negative error are obtained by dividing each numeric value by a correct representative value using rational approximation. Among the obtained irreducible fractions, the one having the smallest common denominator of the numeric values is obtained, which is corrected using the representative value to obtain A.

Abstract: An object of the present invention is to provide a method for encoding signals by separating a signal by using a common multiplier so as to improve the efficiency of compression of audio signals and a method for calculating a common multiplier of a set of numeric values including errors. A common multiplier A, of the input signal samples x(i) in each frame is estimated. Each sample x(i) is divided by A, the quotient is rounded to the number of significant digits, and the rounded quotient y(i) is encoded. The quotient y(i) is multiplied by A and the product is subtracted from x(i) to obtain an error signal z(i), which is then encoded. The multiplier A is encoded and the code of y(i), the code of z(i), and the code of A are combined together and outputted. To estimate the common multiplier, each numeric value is divided by a determined representative value.