$On Lengths of $\mathbb{F}_2[x,y,z]/(x^{d_1}, y^{d_2},z^{d_3}, x+y+z)$$

On Lengths of $\mathbb{F}_2[x,y,z]/(x^{d_1}, y^{d_2},z^{d_3}, x+y+z)$

In this paper, we provide a formula for the vector space dimension of the ring $\mathbb{F}_2[x,y,z]/(x^{d_1}, y^{d_2},z^{d_3}, x+y+z)$ over $\mathbb{F}_2$ when $d_1,d_2,d_3$ all lie between successive powers of $2$. For general $d_1,d_2,d_3$, we provide a simple algorithm to calculate the...

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Veröffentlicht in:	arXiv.org 2024-08
Hauptverfasser:	Han, Fiona, Kenkel, Jennifer, Li, Daniel, Venkatesh, Sridhar, Wiles, Ashley
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Sprache:	eng
Schlagworte:	Algebra Cryptography Field theory Mathematical analysis Polynomials Questions Quotients Rings (mathematics) Vector spaces
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description	In this paper, we provide a formula for the vector space dimension of the ring $\mathbb{F}_2[x,y,z]/(x^{d_1}, y^{d_2},z^{d_3}, x+y+z)$ over $\mathbb{F}_2$ when $d_1,d_2,d_3$ all lie between successive powers of $2$. For general $d_1,d_2,d_3$, we provide a simple algorithm to calculate the vector space dimension of $\mathbb{F}_2[x,y,z]/(x^{d_1}, y^{d_2},z^{d_3}, x+y+z)$ by combining our formula with certain results of Chungsim Han (1992).
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title	On Lengths of $\mathbb{F}_2[x,y,z]/(x^{d_1}, y^{d_2},z^{d_3}, x+y+z)$
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On Lengths of \(\mathbb{F}_2[x,y,z]/(x^{d_1}, y^{d_2},z^{d_3}, x+y+z)\)