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A semifield of order 128 and fractional dimension $\frac{7}{3}$ relative to one of its subsemifields

Ignacio Fernández Rúa, Elías Fernández-Combarro Álvarez

TL;DR

The paper tackles the question of whether semifield planes of order $2^t$ with $(t,3)=1$ can admit semifield subplanes of order $2^3$. It constructs a semifield $S$ of order $2^7$ via a standard basis $\{A_i\}_{i=1}^7$ in $GL(7,2)$, embedding $\mathbb{F}_8$ through the first three blocks associated with the irreducible polynomial $x^3+x+1$ and an $\mathbb{F}_2$-basis $\mathcal B$ that yields a $3$-cube $\mathbf A_{S,\mathcal{B}}$ encoding multiplication. This produces a fractional dimension of $\tfrac{7}{3}$, demonstrating a non-integer dimension case beyond the previously known $m=2$ scenarios. The result expands the catalog of fractional semifields and suggests a path toward families of order $2^n$ with $3 mid n$ that contain $\mathbb{F}_8$, potentially informing the original problem of Chen and Cordero.

Abstract

In this short note, an example of a semifield of order 128 containing the Galois field $\mathbb{F}_8$ is given. Up to our knowledge, this is the first example supporting the following problem by Cordero and Chen (2013): ``There exist semifield planes of order $2^t$, for any integers $t$ relatively prime to 3 that admit semifield subplanes of order $2^3$''.

A semifield of order 128 and fractional dimension $\frac{7}{3}$ relative to one of its subsemifields

TL;DR

The paper tackles the question of whether semifield planes of order with can admit semifield subplanes of order . It constructs a semifield of order via a standard basis in , embedding through the first three blocks associated with the irreducible polynomial and an -basis that yields a -cube encoding multiplication. This produces a fractional dimension of , demonstrating a non-integer dimension case beyond the previously known scenarios. The result expands the catalog of fractional semifields and suggests a path toward families of order with that contain , potentially informing the original problem of Chen and Cordero.

Abstract

In this short note, an example of a semifield of order 128 containing the Galois field is given. Up to our knowledge, this is the first example supporting the following problem by Cordero and Chen (2013): ``There exist semifield planes of order , for any integers relatively prime to 3 that admit semifield subplanes of order ''.

Paper Structure

This paper contains 3 sections, 1 theorem, 5 equations.

Table of Contents

  1. Introduction
  2. The example
  3. Conclusions

Key Result

Proposition 1

Hentzel[Proposition 3] For any finite semifield $S$ of order $2^7$, there exists a set of $7$ matrices, called standard basis, $\{A_i\}_{i=1}^7\subseteq GL(7,2)$, such that: Such a standard basis is the set of coordinate matrix of the $\mathbb{F}_2-$linear maps $\{L_{a_i}(w)=a_iw\}_{i=1}^7$, for a $\mathbb{F}_2-$basis $\mathcal{B}=\{a_1=1,a_2,\dots,a_7\}$ of $S$. In terms of the $3-$cube $\mathbf

Theorems & Definitions (1)

  • Proposition 1