In this paper, a decoding method is proposed for the (41, 21, 9) quadratic residue code. The method is acquired through applying the condition of the modified Gao’s algorithm that is similar to fast transform decoding developed by Shiozaki [8]. The property of the syndrome polynomial is used in our decoding method. The new algorithm for has been verified by a software simulation using C languag...

Regarding non-unique factorization of integer-valued polynomials over a discrete valuation domain (R,M) with finite residue field, it is known that there exist absolutely irreducible elements, is, elements all whose powers factor uniquely, and non-absolutely elements. We completely constructively characterize the among split polynomials. They correspond bijectively to sets, which we call balanc...

We show, in an effective way, that there exists a sequence of congruence classes ak (mod mk) such that the minimal solution n = nk of the congruence φ(n) ≡ ak (mod mk) exists and satisfies log nk/ logmk → ∞ as k → ∞. Here, φ(n) is the Euler function. This answers a question raised in [3]. We also show that every congruence class containing an even integer contains infinitely many values of the ...

We assign to each positive variety V and a fixed natural number k the class of all (positive) boolean combinations of the restricted polynomials, i.e. the languages of the form L0a1L1a2 . . . a`L`, where ` ≤ k, a1, . . . , a` are letters and L0, . . . , L` are from the variety V. For this polynomial operator we give a certain algebraic counterpart which works with identities satisfied by syntac...

We report on an experiment in exploring properties of residue classes over the integers with the combined e ort of a multi-strategy proof planner and two computer algebra systems. An exploration module classies a given set and a given operation in terms of the algebraic structure they form. It then calls the proof planner to prove or refute simple properties of the operation. Moreover, we use d...