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Cyclic Codes

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Part of the Lecture Notes in Mathematics book series (LNM,volume 201)

Abstract

Ir. this chapter R(n) will denote the n-dimensional vector space over GF(q). We shall make the restriction (n,q) = 1. Consider the ring R of all polynomials with coefficients in GF(q), i.e. (GF(q)[x],+,). Let S be the principal ideal in R generated by the polynomial xn − 1, i.e. S := (({xn−1}),+,). R/S is the residue class ring R mod S, i.e. (GF(q)[x] mod ({xn−1},+,). The elements of this ring can be represented by polynomials of degree < n with coefficients in GF(q). The additive group of R/S is isomorphic to R(n). An isomorphism is given by associating the vector a = (a0,a1,...,an−1) with the polynomial a0 + a1x + ... + an−1xn−1.

Keywords

  • Code Word
  • Cyclic Code
  • Minimal Polynomial
  • Principal Ideal
  • Primitive Element

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 1971 Springer-Verlag Berlin Heidelberg

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van Lint, J.H. (1971). Cyclic Codes. In: Coding Theory. Lecture Notes in Mathematics, vol 201. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-20712-3_3

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  • DOI: https://doi.org/10.1007/978-3-662-20712-3_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-05476-4

  • Online ISBN: 978-3-662-20712-3

  • eBook Packages: Springer Book Archive