On isodual double polycirculant codes.
Minjia Shi, Li Xu, Patrick Solé · arXiv (Cornell University) · 2018
Double polycirculant codes are introduced here as a generalization of double circulant codes. They form a special class of quasi-polycyclic codes of index $2.$ 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 can only occur over $ \F_2$ in the double circulant case. Building on the existence of infinitely many irreducible trinomials over $\F_2$ we show that double polycirculant binary codes satisfy the Varshamov-Gilbert bound for linear codes of rate one half. They are thus asymptotically good.