Peter Pal Pach ; Csaba Szabo - On the minimal distance of a polynomial code

dmtcs:556 - Discrete Mathematics & Theoretical Computer Science, November 16, 2011, Vol. 13 no. 4 -
On the minimal distance of a polynomial codeArticle

Authors: Peter Pal Pach 1; Csaba Szabo 1

  • 1 Department of Algebra and Number Theory [Budapest]

For a polynomial f(x) is an element of Z(2)[x] it is natural to consider the near-ring code generated by the polynomials f circle x, f circle x(2) ,..., f circle x(k) as a vectorspace. It is a 19 year old conjecture of Gunter Pilz that for the polynomial f (x) - x(n) broken vertical bar x(n-1) broken vertical bar ... broken vertical bar x the minimal distance of this code is n. The conjecture is equivalent to the following purely number theoretical problem. Let (m) under bar = \1, 2 ,..., m\ and A subset of N be an arbitrary finite subset of N. Show that the number of products that occur odd many times in (n) under bar. A is at least n. Pilz also formulated the conjecture for the special case when A = (k) under bar. We show that for A = (k) under bar the conjecture holds and that the minimal distance of the code is at least n/(log n)(0.223). While proving the case A = (k) under bar we use different number theoretical methods depending on the size of k (respect to n). Furthermore, we apply several estimates on the distribution of primes.

Volume: Vol. 13 no. 4
Published on: November 16, 2011
Accepted on: June 9, 2015
Submitted on: April 28, 2010
Keywords: [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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