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The Weight Spectrum of the Affine Grassmann Code $C^{\mathbb A}(3,6)$
Prasant Singh, Rohit Yadav
TL;DR
The weight spectrum of affine Grassmann codes beyond ℓ=2 was unknown, including C^A(3,6). This paper classifies codewords by the largest minor size in their expressions and determines every possible Hamming weight and the number of codewords at each weight.
Problem
The weight spectrum of affine Grassmann codes beyond ℓ=2 remained unknown, including for C^A(3,6).
Source: §1 · p. 3Method
The authors classify codewords by the largest size of a minor with a nonzero coefficient in their expressions and analyze the resulting cases.
Source: §1 · p. 3Results
Theorem 3.18 gives the Hamming weight and number of codewords for every type, yielding the complete weight distribution of C^A(3,6).
Source: §1 · p. 3, §3.4 · p. 25Takeaways & Limitations
The result settles the first affine Grassmann weight-spectrum case with ℓ≥3.
Source: §3.4 · p. 26The weight-spectrum problem for affine Grassmann codes in general remains open.
Source: §3.4 · p. 26
Abstract
from arXiv · showhide
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 $C^{\mathbb A}(3,6)$.
1 Introduction
The paper determines the complete weight spectrum of the affine Grassmann code C^A(3,6), including the number of codewords attaining each weight. It also presents a streamlined computation of the spectrum of C^A(2,m).
1 Introduction
- Beyond ℓ = 2, affine Grassmann code weight spectra remained unknown, including C^A(3,6) and C^A(3,7); Piñero and Singh determined the ℓ = 2 spectrum using a projection and skew-symmetric matrix classification.They decomposed Grassmann codewords into Schubert-code and affine Grassmann-code components.Source: §1 · pp. 2–3
- Classifying codewords by the largest minor size in their 3 × 3 matrix expressions yields all possible weights of C^A(3,6) and the number of codewords at each weight.The authors use this approach because the natural group action leads to a richer orbit structure, making direct orbit-based analysis more difficult.Source: §1 · p. 3
- The appendix provides a streamlined computation of the weight spectrum of C^A(2,m), improving upon the computation in.Source: §1 · p. 3
2 Preliminaries
The section sets up the affine Grassmann code through evaluation on an affine chart, then introduces coordinate symmetries and counting lemmas used in the weight computation.
2 Preliminaries
- Evaluating minors of a generic ℓ×ℓ′ matrix on the affine chart A^δ(Fq) defines C^A(ℓ,m), with an injective evaluation map and code length q^δ.The chart is an affine open subset of G_{ℓ,m}, and the 0×0 minor is the constant function 1.Source: §2 · pp. 3–4
- The permutations induced by the stated affine transformations preserve C^A(ℓ,m) [BGH12, Lemma 7] and act transitively on its coordinates, enabling same-weight replacement by a more convenient codeword.Source: §2 · p. 4
- Four elementary counting lemmas cover quadratic-equation solutions, affine quadratic zero sets, monic quadratic root counts, and matrices with prescribed determinant.The lemmas are included with proofs for use in the following section.Source: §2 · p. 5
- For monic quadratics, the numbers with exactly 1, 0, and 2 distinct roots are respectively q, q(q−1)/2, and q(q−1)/2; the determinant count's common value for c ≠ 0 is |SL_n(Fq)|.Source: §2 · p. 6
3 Weight Spectrum of CA(3, 6)
The section determines the weight distribution of the affine Grassmann code by classifying codewords according to their highest-degree minor terms and organizing the weight calculations through coefficient ranks and quadratic-root counts. It culminates in the complete distribution, listing 16 distinct Hamming weights and their codeword counts.
3 Weight Spectrum of CA(3, 6)
- Degree-at-most-two codewords are analyzed as constants or affine linear functions, or by quadratic parts of coefficient-matrix ranks one, two, and three.The analysis begins with functions in V0 ⊕ V1 and then classifies quadratic cases using their quadratic parts.Source: §3 · p. 10, §3.1 · pp. 10–12
3.1 Hamming Weights of Linear and Quadratic Functions
- q^9 is the weight of each nonzero constant function, while each nonconstant affine linear function has weight q^9 − q^8; their counts are q − 1 and q^10 − q, respectively.A nonconstant affine linear function has an affine hyperplane of q^8 zeros in F_q^9.Source: §3.1 · p. 11
- For a quadratic part that is a scalar multiple of one 2 × 2 minor, the zero-set size is q^5(q^3 + q^2 − q) when g = k = 0 and q^5(q^3 − q) when g = 0, k ≠ 0.The corresponding cases contain q^4 and q^4(q − 1) codewords, respectively.Source: §3.1 · pp. 11–12
- For this form, if ℓ ≠ 0, |Z(f)| = q^8; if ℓ = 0 and k = 0, |Z(f)| = q^3(q^5 + q^3 − q^2), while if k ≠ 0, |Z(f)| = q^3(q^5 − q^2).Source: §3.1 · p. 13
- For the rank-three quadratic case, the number of roots in F_q of t^2 + bt + d is 0, 1, or 2, and these cases yield three distinct weights.The weight distribution is organized by the root count of this polynomial.Source: §3.1 · pp. 14–15
- For linear and quadratic functions, weight counts are determined by fiber sizes and the number of roots of a quadratic, with the resulting distribution listed in a table.The excerpts do not provide the table’s weight values.Source: §3.1 · p. 15
3.2 Hamming Weights of Functions With Cubic Part
- For cubic functions det(X) + ℓ + α0, the linear part is reduced by rank to four partial-trace cases, and the trace case depends on roots of t^2 + α0t − 1.For α0 = 0, the root count is two when q is odd and one when q is even; for nonzero α0, the passage defines the root count as Nf.Source: §3.2 · pp. 16, 19
- The Hamming weight has a stated formula for f=det(X)+α0 when ℓ=0, for det(X)+X11+α0, and for det(X)+X11+X22+α0; in the second case, all q−1 nonzero α0 values give the same weight.Source: §3.2 · pp. 16, 18
3.3 Automorphisms and Stabilizers
- For det X + ℓ(X) + α0, every stabilizer has translation U = 0, and the stabilizer is unchanged by shifting α0; its form is constrained by QESP = ES and det P det Q = 1.For ℓ = tr X, the stabilizer size is |GL3(Fq)|.Source: §3.3 · pp. 20–21
3.4 The Complete Weight Distribution
- The combined theorem covers all q^20 codewords, with rank counts including ρ2 = q(q −1)^2(q + 1)(q^2 + q + 1)^2 and ρ3 = |GL3(Fq)|.The theorem is introduced as combining the preceding results; its full weight distribution is not included in these passages.Source: §3.4 · p. 22
- For even q, the quadratic-root counts are µ0=q/2, µ1=0, and µ2=q/2−1; for odd q, µ1=1+ε and µ2=(q−4−ε)/2.For odd q, ε is determined by whether −1 is a square in Fq; these counts classify nonzero constants by the number of roots of t^2+α0t−1.Source: §3.4 · pp. 24–25
- C^A(3,6) has 16 distinct Hamming weights; Table 1 gives the codeword count and representative polynomial type for each, and the distribution totals q^20 codewords.The theorem states that every codeword belongs to exactly one listed type; the supplied table caption does not expose the individual weight or count values.Source: §3.4 · pp. 25–26, Table 1 · p. 27
- For codewords with a nonzero determinant term, weights are organized by the rank of the linear part and whether the constant is zero; full-rank nonzero-constant cases split by quadratic-root count.Translation reduces the quadratic-plus-linear-plus-constant part to a determinant-plus-linear-plus-constant form, with q^9 codewords per resulting pair; a nonzero determinant coefficient contributes an additional factor q−1 when counting.Source: §3.4 · pp. 24, 26
- The general affine Grassmann weight spectrum remains open; this case has 15 nonzero weights versus 5 for C(3,6), and the method suggests induction on ℓ.The discussion attributes the difficulty to classifying forms, the finer affine-cell symmetries, and field arithmetic entering through quadratic-root counts.Source: §3.4 · p. 26