Complete list of codeword weights found in special affine Grassmann code
The Weight Spectrum of the Affine Grassmann Code $C^{\mathbb A}(3,6)$
Information Theory
Summary
This paper studies a specific kind of error-correcting code called the affine Grassmann code constructed from a geometric object related to matrices. The authors represent codewords using certain building blocks called minors, which are determinants of parts of a matrix. They group these codewords by the size of the biggest minor involved and use this to find all possible numbers of nonzero entries (weights) that codewords can have. They also count how many codewords have each possible weight, giving a full overview of the code's weight spectrum.
affine Grassmann codeGrassmanniancodewordsHamming weightminorsdeterminanterror-correcting codeweight spectrumlinear combinationmatrix
Authors
Prasant Singh, Rohit Yadav
Abstract
In this article, we consider the affine Grassmann code $C^{\mathbb A}(3,6)$, obtained from the affine open cell ${\mathbb A}^9$ of the Grassmannian $G_{3,6}$. We exploit the representation of codewords as linear combinations of minors of all sizes of a generic $3\times3$ matrix and classify them according to the largest size of a minor occurring with a nonzero coefficient. Using this classification, we determine all possible Hamming weights of codewords of $C^{\mathbb A}(3,6)$ and, for each weight, compute the number of codewords attaining that weight. Consequently, we obtain the complete weight spectrum of the affine Grassmann code $C^{\mathbb A}(3,6)$.