The Existence For Several Types Of Near-Perfect Numbers And Self-Orthogonal Cyclic Codes Over Finite Fields

Abstrct Let a be a positive integer, p1and p2be two distinct odd primes with p1<P2. In this paper, by elementary methods and techniques, we give a necessary and sufficient condition for that n=2αp12p2or2αp1p2is an even near-perfect number. We also give a new proof for Theorem1in [35], and then obtain an equivalent condition for that the κ-near-perfect number in the form n=2αp is also a near-perfect number, where κ≥2, α+1> r1> r2>> rκ≥1, and p=2α+1-2r1-…-2rκ-1is an odd prime. On the other hand, we also obtain a necessary and sufficient condition for that there exists a non-trivial self-orthogonal or self-dual cyclic code over finite fields and the explicit enumerating formula. As a corollary, a simple and efficient criterion for the existence of non-trivial self-orthogonal cyclic codes are obtained.