https://hal.inria.fr/hal-00990488Pach, Peter PalPeter PalPachDepartment of Algebra and Number Theory [Budapest] - ELTE - Eötvös Loránd UniversitySzabo, CsabaCsabaSzaboDepartment of Algebra and Number Theory [Budapest] - ELTE - Eötvös Loránd UniversityOn the minimal distance of a polynomial codeHAL CCSD2011[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]Inria Sophia Antipolis-Méditerranée / I3s, Service Ist2014-05-13 15:39:192017-09-07 01:03:392014-05-13 16:25:19enJournal articleshttps://hal.inria.fr/hal-00990488/document10.46298/dmtcs.556application/pdf1For 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.