Skip to main content
Springer Nature Link
Log in

An enumeration formula for certain irreducible polynomials with an application to the construction of irreducible polynomials over the binary field

  • Published:
Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract

We establish a formula for the number of irreducible polynomialsf(x) over the binary fieldF 2 of given degreen ≧ 2 for which the coefficient ofx n-1 and ofx is equal to 1. This formula shows that the number of such polynomials is positive for alln ≧ 2 withn ≠ 3. These polynomials can be applied in a construction of irreducible self-reciprocal polynomials overF 2 of arbitrarily large degrees.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

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

    Google Scholar 

  2. Hayes, D. R.: The distribution of irreducibles inGF[q,x]. Trans. Am. Math. Soc.117, 101–127 (1965)

    Google Scholar 

  3. Lidl, R., Niederreiter, H.: Finite fields. Reading, MA.: Addison—Wesley 1983

    Google Scholar 

  4. Lidl, R., Niederreiter, H.: Introduction to finite fields and their applications. Cambridge: Cambridge University Press 1986

    Google Scholar 

  5. Meyn, H.: On the construction of irreducible self-reciprocal polynomials over finite fields. AAECC1, 43–53 (1990)

    Google Scholar 

  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)

    Google Scholar 

  7. Wiedemann, D.: An iterated quadratic extension ofGF(2). Fibonacci Quart.26, 290–295 (1988)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Information Processing, Austrian Academy of Sciences, Sonnenfelsgasse 19, A-1010, Vienna, Austria

    Harald Niederreiter

Authors
  1. Harald Niederreiter
    View author publications

    You can also search for this author inPubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Niederreiter, H. An enumeration formula for certain irreducible polynomials with an application to the construction of irreducible polynomials over the binary field. AAECC 1, 119–124 (1990). https://doi.org/10.1007/BF01810295

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01810295

Keywords