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Applicable Algebra in Engineering, Communication and Computing

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02-01-2019 | Original Paper

A Swan-like note for a family of binary pentanomials

Author: Giorgos Kapetanakis

Published in: Applicable Algebra in Engineering, Communication and Computing | Issue 5/2019

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Abstract

In this note, we employ the techniques of Swan (Pac J Math 12(3):1099–1106, 1962) with the purpose of studying the parity of the number of the irreducible factors of the penatomial \(X^n+X^{3s}+X^{2s}+X^{s}+1\in \mathbb {F}_2[X]\), where s is even and \(n>3s\). Our results imply that if \(n \not \equiv \pm 1 \pmod {8}\), then the polynomial in question is reducible.

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As a confirmation of our results, we verified our results with Magma for \(7\le n\le 1000\) and found identical results.

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Title: A Swan-like note for a family of binary pentanomials
Author: Giorgos Kapetanakis
Publication date: 02-01-2019
Publisher: Springer Berlin Heidelberg
Published in: Applicable Algebra in Engineering, Communication and Computing / Issue 5/2019
Print ISSN: 0938-1279
Electronic ISSN: 1432-0622
DOI: https://doi.org/10.1007/s00200-018-0378-7

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