Codes and finite geometries

We explore the connections between finite geometry and algebraic coding theory, giving a rather full account of the Reed-Muller and generalized Reed-Muller codes. Some of the results and many of the proofs are new but this is largely an expository effort that relies heavily on the work of Delsarte et al. and of Charpin. The necessary geometric background is sketched before we begin the discussion of the Reed-Muller codes and their p-ary analogues. We prove all the classical results concerning these codes and include a discussion of the group-algebra approach and prove Berman's theorem characterizing the codes as powers of the radical. Included also is a discussion of the characterization of affine-invariant cyclic codes given by Kasami, Lin and Peterson and its generalization by Delsarte. our theme throughout this work is the relationship between these codes and the codes coming from both affine and projective geometries. The final section develops the theory in the more difficult case in which the field is not of prime order, here must look at subfield subcodes - which complicates the connection with the geometric codes, which are codesover the prime subfield of the field of the geometry

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

Symmetries of weight enumerators and applications to Reed-Muller codes

Gleason's 1970 theorem on weight enumerators of self-dual codes has played a crucial role for research in coding theory during the last four decades. Plenty of generalizations have been proved but, to our knowledge, they are all based on the symmetries given by MacWilliams' identities. This paper is intended to be a first step towards a more general investigation of symmetries of weight enumerators. We list the possible groups of symmetries, dealing both with the finite and infinite case, we develop a new algorithm to compute the group of symmetries of a given weight enumerator and apply these methods to the family of Reed-Muller codes, giving, in the binary case, an analogue of Gleason's theorem for all parameters.Comment: 14 pages. Improved and extended version of arXiv:1511.00803. To appear in Advances in Mathematics of Communication

Hypersurfaces in weighted projective spaces over finite fields with applications to coding theory

We consider the question of determining the maximum number of $\mathbb{F}_q$-rational points that can lie on a hypersurface of a given degree in a weighted projective space over the finite field $\mathbb{F}_q$, or in other words, the maximum number of zeros that a weighted homogeneous polynomial of a given degree can have in the corresponding weighted projective space over $\mathbb{F}_q$. In the case of classical projective spaces, this question has been answered by J.-P. Serre. In the case of weighted projective spaces, we give some conjectures and partial results. Applications to coding theory are included and an appendix providing a brief compendium of results about weighted projective spaces is also included