Codes on Weighted Projective Spaces and Their Algebraic Invariants

Weighted projective spaces are natural generalizations of classical projective spaces having rich structures and exhibiting interesting algebraic geometric properties. They have been regarded as convenient ambient spaces to create interesting classes of linear codes over finite fields in the literature (see, [1],[2],[3]). Weighted Projective Reed–Muller codes were introduced by Sørensen in [3].
The purpose of this talk is to introduce these codes over a finite field, to give results for their main parameters, and to reveal the role of computer algebra packages to study some of the relevant combinatorial commutative algebraic invariants. We pay a particular attention on two dimensional case to obtain more explicit information about the minimal free resolution of the vanishing ideal of the weighted projective plane $\mathbb{P}(1, a, b)$ over $\mathbb{F}_q$ . This yields to the Hilbert function giving the dimension of the code and regularity index which is crucial to eliminate trivial codes. We also compute the minimum distance of codes on the weighted projective space of the form $\mathbb{P}(1, 1, a)$.

[1] Y. AUBRY , W. CASTRYCK , S. R. GHORPADE , G. LACHAUD , M. E. O’SULLIVAN , AND S. RAM.,
Hypersurfaces in weighted projective spaces over finite fields with applications to coding theory. In Algebraic
Geometry for Coding Theory and Cryptography, Springer, (2017)
[2] O. GEIL, C. THOMSEN , Weighted Reed–Muller codes revisited. Desing Codes and Cryptography, (2013)
[3] A. B. SORENSEN , Weighted Reed–Muller codes and algebraic-geometric codes. IEEE Trans.Inf.Theory, (1992)<>