×

An enumeration formula for certain irreducible polynomials with an application to the construction of irreducible polynomials over the binary field. (English) Zbl 0726.11077

By evaluating some Kloosterman sums the author obtains a formula for the number of irreducible polynomials of degree n over GF(2) such that the coefficients of x and \(x^{n-1}\) are 1. From this formula it readily can be seen that this number is always positive. Therefore such polynomials exist for arbitrarily large degrees. They can be used in the construction of irreducible self-reciprocal polynomials over GF(2).

MSC:

11T06 Polynomials over finite fields
11T24 Other character sums and Gauss sums
12E05 Polynomials in general fields (irreducibility, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Brawley, J. V., Schnibben, G. E.: Infinite algebraic extensions of finite fields. Contemporary Math., Vol. 95. Providence, R. I.: American Math. Society 1989 · Zbl 0674.12009
[2] Hayes, D. R.: The distribution of irreducibles inGF[q,x]. Trans. Am. Math. Soc.117, 101–127 (1965) · Zbl 0139.27502
[3] Lidl, R., Niederreiter, H.: Finite fields. Reading, MA.: Addison–Wesley 1983 · Zbl 0554.12010
[4] Lidl, R., Niederreiter, H.: Introduction to finite fields and their applications. Cambridge: Cambridge University Press 1986 · Zbl 0629.12016
[5] Meyn, H.: On the construction of irreducible self-reciprocal polynomials over finite fields. AAECC1, 43–53 (1990) · Zbl 0724.11062 · doi:10.1007/BF01810846
[6] Varshamov, R. R., Garakov, G. A.: On the theory of selfdual polynomials over a Galois field (Russian). Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.)13, 403–415 (1969) · Zbl 0228.12003
[7] Wiedemann, D.: An iterated quadratic extension ofGF(2). Fibonacci Quart.26, 290–295 (1988) · Zbl 0658.12012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.