In this manuscript, we study affine polar Grassmann codes. These are linear codes associated to an affine part of the projective embedding of a polar Grassmannian. In particular, the length, dimension, and minimum distance of the affine polar Grassmann codes derived from the symplectic Grassmannian and the Hermitian Grassmannian of totally isotropic spaces are determined. It is also shown that affine Hermitian Grassmann codes have a better minimum distance than affine Grassmann codes while having the same length, dimension, and alphabet size. We also study their dual codes and their related incidence structure.

In this article, we consider decoding Grassmann codes, linear codes associated to the Grassmannian and its embedding in a projective space. We look at the orbit structure of Grassmannian arising from the multiplicative group${\mathbb {F}}_{q^{m}}^{*}$in$GL_{m}(q)$. We project the corresponding Grassmann code onto these orbits to obtain a subcode of a$q$–ary Reed-Solomon code. We prove that some of these projections contain an information set of the parent Grassmann code. By improving the decoding capacity of Peterson’s decoding algorithm for the projected subcodes, we prove that one can correct up to$\lfloor (d-1)/2\rfloor $errors for Grassmann code, where$d$is the minimum distance of Grassmann code.

In this paper, we construct codes for local recovery of erasures with high availability and constant-bounded rate from the Hermitian curve. These new codes, called Hermitian-lifted codes, are evaluation codes with evaluation set being the set of $\mathbb{F}_{q^2}$-rational points on the affine curve. The novelty is in terms of the functions to be evaluated; they are a special set of monomials which restrict to low degree polynomials on lines intersected with the Hermitian curve. As a result, the positions corresponding to points on any line through a given point act as a recovery set for the position corresponding to that point.

In this paper, we prove that Schubert union codes are Tanner codes constructed from the point-line incidence geometry inherited from the Grassmannian. Our proof is based on an iterative encoding algorithm for Tanner codes. This encoder determines the entries of a code word of a Tanner code from the entries in a given subset of its positions. As a result, we find sufficient conditions on the initial positions such that a code word is determined from the component codes only. This algorithm has linear complexity in the code length. We also use this encoder to determine the minimum distance of Schubert union codes in terms of the minimum distance of the Schubert varieties contained therein.

We study a class of graph based codes with Reed-Solomon component codes as affine variety codes. We give a formulation of the exact dimension of graph codes in general. We give an algebraic description of these codes which makes the exact computation of the dimension of the graph codes easier.