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Revisiting the Weight Spectrum of the Affine Grassmann Code $C^{\mathbb{A}}(2,m)$
Rohit Yadav
TL;DR
For affine Grassmann codes, the spectrum is known for ℓ=2 and CA(3,6); Piñero and Singh determined the ℓ=2 spectrum by case analysis on alternating-matrix rank. The paper expresses codeword weights through a single closed formula involving two explicit affine subspaces and their intersection. The independent derivation works for every prime power q, with zero and nonzero constant codewords included as degenerate instances.
Problem
For affine Grassmann codes, the spectrum is known for ℓ=2 and CA(3,6); Piñero and Singh determined the ℓ=2 spectrum by case analysis on alternating-matrix rank.
Source: §1 · p. 2Method
The paper expresses codeword weights through a single closed formula involving two explicit affine subspaces and their intersection.
Source: §1 · p. 2Results
The independent derivation works for every prime power q, with zero and nonzero constant codewords included as degenerate instances.
Source: §1 · p. 2Takeaways & Limitations
The uniform treatment is also useful beyond ℓ=2, supplying a key ingredient for a reduction when ℓ=3.
Source: §1 · pp. 2–3The weight-enumeration setup fixes ℓ=2 and m=2+ℓ′, with ℓ′≥2.
Source: §2 · p. 3
Abstract
from arXiv · showhide
Affine Grassmann codes, introduced by Beelen, Ghorpade and H{\o}holdt, 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 Pi\~nero 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.
1 Introduction
Affine Grassmann codes evaluate linear combinations of generic-matrix minors, extending evaluation-code constructions. For C^A(2,m), this paper replaces an earlier rank-case analysis with a unified weight formula, uniform over prime powers.
1 Introduction
- Affine Grassmann codes evaluate linear combinations of generic-matrix minors at every point of the corresponding affine space over Fq, extending evaluation-code constructions.For ℓ≥2, higher-order minors distinguish them from first-order Reed–Muller codes.Source: §1 · pp. 1–2
- Earlier work studied the dual code, automorphisms, generalized Hamming weights, and majority-logic decoding.The codes were first studied by Beelen, Ghorpade and Høholdt.Source: §1 · p. 2
- Piñero and Singh determined the C^A(2,m) weight spectrum by analyzing the rank of an associated alternating matrix; spectra for ℓ≥3 remain largely open.The spectrum also determines the dual spectrum via MacWilliams identities and governs memoryless-channel decoding-error probability.Source: §1 · p. 2
- A single closed formula expresses C^A(2,m) codeword weights using two explicit affine subspaces and their intersection, with zero and nonzero constant codewords included as degenerate cases.The derivation works uniformly over every prime power, including characteristic two.Source: §1 · p. 2
- Specializing the formula supplies a key ingredient in a reduction from C^A(ℓ,m) to C^A(ℓ−1,m−1) when ℓ=3.The paper identifies the uniform treatment as useful beyond ℓ=2.Source: §1 · pp. 2–3
2 Weight Enumeration of CA(2, m)
For ℓ=2, the paper parameterizes codewords and expresses their weights through point counts on two affine subspaces and their intersection. It then uses the possible ranks and intersections to enumerate weights and multiplicities.
Codeword parameterization
- For ℓ=2, each codeword has a unique expression using a constant, linear entries, and 2×2 minors, encoded by scalar, vector, and alternating-matrix parameters.The matrix B is alternating, with zero diagonal, including in characteristic two.Source: §2 · pp. 3–4
Weight computation
- For fixed u, f(u,v) is affine in v; its linear part vanishes on U, while its constant term vanishes on V.This split determines how many nonzero evaluations occur for each u.Source: §2 · pp. 4, 6
Lemma 2.1
- If rank B = 2r, nonempty U is a coset of ker B of size q^(ℓ′−2r), while M is 0, q^(ℓ′−2r−1), or q^(ℓ′−2r).The largest value of M occurs only when U0 is contained in V0.Source: §2 · pp. 4–5
Theorem 2.2
- The weight formula is wt(cf) = q^(2ℓ′) − q^(2ℓ′−1) + Nq^(ℓ′−1) − Mq^ℓ′, where N = #U and M = #(U ∩ V).The formula reduces the weight computation to the sizes of the two affine subspaces and their intersection.Source: §2 · pp. 5–6
- For rank B = 2r, wt(cf) takes one of three values: q^(2ℓ′) − q^(2ℓ′−1) + q^(2ℓ′−2r−1), q^(2ℓ′) − q^(2ℓ′−1), or q^(2ℓ′) − q^(2ℓ′−1) + q^(2ℓ′−2r−1) − q^(2ℓ′−2r).Source: §2 · p. 6
Theorem 2.5
- Theorem 2.5 combines the weight formula with matrix counts to give the complete distribution; the listed weights are pairwise distinct and exhaustive.The zero word has multiplicity 1, and the maximum weight q^(2ℓ′) has multiplicity q − 1.Source: §2 · p. 7
Remark 2.7
- CA(2,m) has 2⌊m/2⌋+1 distinct weights, versus ⌊m/2⌋+1 for the Grassmann code C(2,m).The formula is also used in to determine the spectrum of CA(3,6).Source: §2 · pp. 8–9