112 research outputs found
Construction of isodual codes from polycirculant matrices
Double polycirculant codes are introduced here as a generalization of double
circulant codes. When the matrix of the polyshift is a companion matrix of a
trinomial, we show that such a code is isodual, hence formally self-dual.
Numerical examples show that the codes constructed have optimal or
quasi-optimal parameters amongst formally self-dual codes. Self-duality, the
trivial case of isoduality, can only occur over \F_2 in the double circulant
case. Building on an explicit infinite sequence of irreducible trinomials over
\F_2, we show that binary double polycirculant codes are asymptotically good
On the proximity of large primes
By a sphere-packing argument, we show that there are infinitely many pairs of
primes that are close to each other for some metrics on the integers. In
particular, for any numeration basis , we show that there are infinitely
many pairs of primes the base expansion of which differ in at most two
digits. Likewise, for any fixed integer there are infinitely many pairs of
primes, the first digits of which are the same. In another direction, we
show that, there is a constant depending on such that for infinitely
many integers there are at least primes which differ from
by at most one base digit
Skew Cyclic codes over \F_q+u\F_q+v\F_q+uv\F_q
In this paper, we study skew cyclic codes over the ring
R=\F_q+u\F_q+v\F_q+uv\F_q, where , and
is an odd prime. We investigate the structural properties of skew cyclic codes
over through a decomposition theorem. Furthermore, we give a formula for
the number of skew cyclic codes of length over $R.
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