The Weight Spectrum of the Affine Grassmann Code $C^{\mathbb{A}}(3,6)$

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 $3\times 3$ 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 $C^{\mathbb{A}}(3,6)$.

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