Decoding of Projective Reed-Muller Codes by Dividing a Projective Space into Affine Spaces

A projective Reed-Muller (PRM) code, obtained by modifying a (classical) Reed-Muller code with respect to a projective space, is a doubly extended Reed-Solomon code when the dimension of the related projective space is equal to 1. The minimum distance and dual code of a PRM code are known, and some decoding examples have been represented for low-dimensional projective space. In this study, we construct a decoding algorithm for all PRM codes by dividing a projective space into a union of affine spaces. In addition, we determine the computational complexity and the number of errors correctable of our algorithm. Finally, we compare the codeword error rate of our algorithm with that of minimum distance decoding.Comment: 17 pages, 4 figure

The Freeness and Minimal Free Resolutions of Modules of Differential Operators of a Generic Hyperplane Arrangement

Let A be a generic hyperplane arrangement composed of r hyperplanes in an n-dimensional vector space, and S the polynomial ring in n variables. We consider the S-submodule D(m)(A) of the nth Weyl algebra of homogeneous differential operators of order m preserving the defining ideal of A. We prove that if n \geq 3, r > n,m > r - n + 1, then D(m)(A) is free (Holm's conjecture). Combining this with some results by Holm, we see that D(m)(A) is free unless n \geq 3, r > n,m < r - n + 1. In the remaining case, we construct a minimal free resolution of D(m)(A) by generalizing Yuzvinsky's construction for m = 1. In addition, we construct a minimal free resolution of the transpose of the m-jet module, which generalizes a result by Rose and Terao for m = 1.Comment: 20 page

Freeness for restriction arrangements of the extended Shi and Catalan arrangements

The extended Shi and Catalan arrangements are well investigated arrangements. In this paper, we prove that the cone of the extended Catalan arrangement of type A is always hereditarily free, while we determine the dimension that the cone of the extended Shi arrangement of type A is hereditarily free. For this purpose, using digraphs, we define a class of arrangements which is closed under restriction and which contains the extended Shi and Catalan arrangements. We also characterize the freeness for the cone of this arrangement by graphical conditions.Comment: 18 pages, 8 figure

Bases for modules of differential operators

It is well-known that the derivation modules of Coxeter arrangements are free. Holm began to study the freeness of modules of differential operators on hyperplane arrangements. In this paper, we study the cases of the Coxter arrangements of type A, B and D. In this case, we prove that the modules of differential operators of order 2 are free. We give examples of all the 3-dimensional classical Coxeter arrangements. Two keys for the proof are ``Cauchy–Sylvester's theorem on compound determinants'' and ``Saito–Holm's criterion''