Showing papers on "Coprime integers published in 1971"

Cyclic and multiresidue codes for arithmetic operations

Abstract: In this paper, the cyclic nature of AN codes is defined after a brief summary of previous work in this area is given. New results are shown in the determination of the range for single-error-correcting AN codes when A is the product of two odd primes p_1 and p_2 , given the orders of 2 modulo p_1 and modulo p_2 . The second part of the paper treats a more practical class of arithmetic codes known as separate codes. A generalized separate code, called a multiresidue code, is one in which a number N is represented as \begin{equation} [N, \mid N \mid _ {m1}, \mid N \mid _{m2}, \cdots , \mid N \mid _{mk}] \end{equation} where m_i are pairwise relatively prime integers. For each AN code, where A is composite, a multiresidue code can be derived having error-correction properties analogous to those of the AN code. Under certain natural constraints, multiresidue codes of large distance and large range (i.e., large values of N ) can be implemented. This leads to possible realization of practical single and/or multiple-error-correcting arithmetic units.

Single- and multiple-burst-correcting properties of a class of cyclic product codes

Abstract: The direct product of p single parity-check codes of block lengths n_1,n_2, \cdots ,n_p is a cyclic code of block length n_1 \times n_2 \times \cdots \times n_p with (n_1 - 1) \times (n_2 - 1) \times \cdots \times (n_p - 1) information symbols per block, if the integers n_1,n_2 \cdots ,n_p are relatively prime in pairs. A lower bound for the single-burst-correction (SBC) capability of these codes is obtained. Then, a detailed analysis is made for p = 3 , and it is shown that the codes can correct one long burst or two short bursts of errors. A lower bound for the double-burst-correction (DBC) capability is derived, and a simple decoding algorithm is obtained. The generalization to correcting an arbitrary number of bursts is discussed.

On the Feit-Thompson Conjecture

Abstract: A counterexample is found to the conjecture that (pw - 1)/@ - 1) and (qP - 1)1(q- 1) are coprime where p, q are primes.

Minimax approximations subject to a constraint

Abstract: A class of approximation problems is considered in which a continuous, positive function So(x) is approximated by a rational function satisfying some identity. It is proved under certain hypotheses that there is a unique rational approximation satisfying the constraint and yielding minimax relative error and that the corresponding relative- error function does not have an equal-ripple graph. This approximation is, moreover, just the rational approximation to q'(x) yielding minimax logarithmic error. This approximation, in turn, is just a constant multiple of the rational approximation to So(x) yielding minimax relative error but not necessarily satisfying the constraint. 1. Introduction. Various authors have investigated approximation problems in which the approximation f(x) is required to satisfy some functional constraint. For example, Cody and Ralston (1) investigated the problem of finding a rational function f(x) with numerator and denominator of degree N such that f(x) satisfies the constraint In this paper, we consider a class of approximation problems including the Cody- Ralston problem and similar problems that have arisen in other contexts. We show that for a problem in this class there is a unique approximation optimal in the sense that it yields minimax relative error, and we characterize this solution. 2. Relative and Logarithmic Error. Suppose that we want to find a polynomial or rational approximation for a function sp(x) on an interval I: a < x < b, where so(x) is continuous and does not vanish in L Then, we may assume that pO(x) is positive for x in I. Let V be a set of admissible functions. Here V will be either the set of all poly- nomials of degree ? M or else the set V will be the set of all rational functions p(x)/q(x) wherep(x) and q(x) are relatively prime polynomials of degree ? M and < N, respectively, and q(x) does not vanish for x in I. We shall refer to such functions p(x)/q(x) as (M, N) rational functions. For f(x) in V, we set R(x) - f(x) - p(x) O(x)