On the minimal distance of a polynomial code
Péter Pál Pach, Csaba Szabó
Abstract
For a polynomial f(x)∈ℤ2[x] it is
natural to consider the near-ring code generated by the polynomials
fˆx,fˆx2, … , fˆxk
as a vectorspace. It is a 19 year old conjecture of
Günter Pilz that for the polynomial f(x)=
xn+xn-1+⋯+x the minimal distance of
this code is n. The conjecture is equivalent to the
following purely number theoretical problem. Let
m={1,2, … ,m} and
A⊂ℕ be an arbitrary finite subset of
ℕ. Show that the number of products that occur odd
many times in n·A is at
least n. Pilz also formulated the conjecture for the
special case when A=k. We show
that for A=k 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 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.
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