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 $q$, we show that there are infinitely many pairs of primes the base $q$ expansion of which differ in at most two digits. Likewise, for any fixed integer $t,$ there are infinitely many pairs of primes, the first $t$ digits of which are the same. In another direction, we show that, there is a constant $c$ depending on $q$ such that for infinitely many integers $m$ there are at least $c\log \log m$ primes which differ from $m$ by at most one base $q$ digit

Constructions of optimal LCD codes over large finite fields

In this paper, we prove existence of optimal complementary dual codes (LCD codes) over large finite fields. We also give methods to generate orthogonal matrices over finite fields and then apply them to construct LCD codes. Construction methods include random sampling in the orthogonal group, code extension, matrix product codes and projection over a self-dual basis.Comment: This paper was presented in part at the International Conference on Coding, Cryptography and Related Topics April 7-10, 2017, Shandong, Chin

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 $u^{2}=u,v^{2}=v,uv=vu$, $q=p^{m}$ and $p$ is an odd prime. We investigate the structural properties of skew cyclic codes over $R$ through a decomposition theorem. Furthermore, we give a formula for the number of skew cyclic codes of length $n$ over $R.