Codes from Veronese and Segre Embeddings and Hamada's Formula

Abstract

In this article we study the codes given by l hypersurfaces in P n q to obtain a new formula for the dimension of codes given by (n&l) flats. We also obtain a new formula for the dimension of the & th order generalized Reed Muller code and describe the code given by the hyperplane intersections of the Segre embedding of P n q _P m q .

Pseudo-embeddings of the (point, k-spaces)-geometry of PG(n, 2) and projective embeddings of DW(2n − 1, 2)

“…This is precisely Corollary 5.3.2 of Assmus and Key [1]. General formulas for the dimension of the pary code of the points and k-dimensional subspaces of PG(n, q), q = p h with p prime and h ∈ N \ {0}, can be found in Hamada [17,Theorem 1] or Inamdar and Sastry [19,Theorem 2.13]. These formulas are usually more complex.…”

“…These formulas are usually more complex. E.g., Theorem 2.13 of [19] tells us that the dimension of the above-mentioned binary code is also equal to 1 + n−k Let ∆ = DW (2n − 1, 2) with n ∈ N \ {0, 1} denote the symplectic dual polar space associated with a symplectic polarity of PG(2n − 1, 2). Let x denote a point of ∆.…”

The permutation action of finite symplectic groups of odd characteristic on their standard modules

“…The incidence matrices between P and flats of PG(2m − 1, q) have been studied extensively over the past forty years. See for example, [2,[5][6][7]15] for F p -ranks of these matrices, and [3,13] for their Smith normal forms. The spaces k[P ] and k[V ] of k-valued functions on P and V , respectively, are permutation modules for the general linear group GL(V ) and were investigated from this viewpoint in [2], where the p-ranks of the above incidence matrices were obtained as a corollary of the description of the submodule lattice of k[P ] (see [2]).…”

“…The incidence matrices between P and flats of PG(2m − 1, q) have been studied extensively over the past forty years. See for example, [14,6,5,1,7] for F pranks of these matrices, and [12,2] for their Smith normal forms. The study of p-ranks of these incidence matrices led the authors of [1] to investigate the submodule lattices of the spaces k[P ] and k[V ] of k-valued functions on P and V respectively, viewed as permutation modules for the general linear group GL(V ).…”

Segre embedding and related maps and immersions in differential geometry

“…It is well-known that the Segre embedding plays an important role in algebraic geometry (see [41,53]). The Segre embedding has also been applied to differential geometry as well as to coding theory (see, for instance, [36,49,52]) and to mathematical physics (see, for instance, [4,51]).…”