Complete weight spectrum found for affine Grassmann code C A 3 6
The Weight Spectrum of the Affine Grassmann Code $C^{\mathbb A}(3,6)$
Information Theory
Summary
The paper studies a specific type of error-correcting code called the affine Grassmann code C^A(3,6), which comes from a mathematical object known as the Grassmannian. The authors classify the codewords by the size of the matrix minors they contain and use this to find all possible numbers of errors the code can detect or correct. They also count how many codewords have each possible error-correcting strength, providing a full description of the code's error properties. This helps understand the code's behavior in detecting and fixing errors in data.
What this means in practice
- •For coding engineers: Design error-correcting codes with precisely known error detection and correction capabilities based on the complete weight spectrum of C^A(3,6).
- •For data transmission teams: Select and implement affine Grassmann codes with predictable error rates for more reliable data communication systems.
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 $C^{\mathbb A}(3,6)$.