Construction of isodual codes from polycirculant matrices
This work addresses the need for efficient code constructions in coding theory, particularly for formally self-dual codes, but it is incremental as it builds on existing double circulant codes.
The paper tackles the problem of constructing isodual codes by introducing double polycirculant codes as a generalization of double circulant codes, showing that these codes are isodual and formally self-dual when using trinomial companion matrices, with numerical examples indicating optimal or quasi-optimal parameters among formally self-dual codes.
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.