Streamlined formula gives precise codeword weights in affine grassmann codes
Revisiting the Weight Spectrum of the Affine Grassmann Code $C^{\mathbb{A}}(2,m)$
Information Theory
Summary
Affine Grassmann codes are special tools used in coding theory, where data is transformed using mathematical matrices. Finding the 'weight spectrum' means understanding how codewords vary in a measurable way called Hamming weight, which affects error correction. Previously, this involved complicated case-by-case analysis based on matrix properties. The authors provide a simpler, unified formula that expresses these weights clearly for a broad range of parameters, making it easier to understand and work with these codes.
What this means in practice
- •For communication engineers: Calculate error-correcting capabilities of codes based on affine Grassmann codes using a simpler formula for codeword weights.
- •For coding algorithm developers: Implement efficient software modules to simulate and analyze affine Grassmann codes with exact weight computations for performance testing.
Authors
Rohit Yadav
Abstract
Affine Grassmann codes, introduced by Beelen, Ghorpade and Høholdt, are linear codes over $\mathbb{F}_q$ obtained by evaluating linear combinations of minors of a generic matrix. The weight spectrum of the affine Grassmann code $C^{\mathbb{A}}(2,m)$ was determined by Piñero and Singh via a case analysis on the rank of an associated alternating matrix. We give an independent and more streamlined derivation, valid for all $m\ge4$ and every prime power $q$, in which each codeword is written in a compact matrix form and its Hamming weight is expressed through a single closed formula involving two explicit affine subspaces and their intersection.