HAL will be down for maintenance from Friday, June 10 at 4pm through Monday, June 13 at 9am. More information
Skip to Main content Skip to Navigation
Journal articles

On the minimal distance of a polynomial code

Abstract : 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.
Document type :
Journal articles
Complete list of metadata

Cited literature [7 references]  Display  Hide  Download

Contributor : Service Ist Inria Sophia Antipolis-Méditerranée / I3s Connect in order to contact the contributor
Submitted on : Tuesday, May 13, 2014 - 3:39:19 PM
Last modification on : Thursday, September 7, 2017 - 1:03:39 AM
Long-term archiving on: : Monday, April 10, 2017 - 10:14:15 PM


Files produced by the author(s)




Peter Pal Pach, Csaba Szabo. On the minimal distance of a polynomial code. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2011, Vol. 13 no. 4 (4), pp.33--43. ⟨10.46298/dmtcs.556⟩. ⟨hal-00990488⟩



Record views


Files downloads