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Menichetti's nonassociative $G$-crossed product algebras and Menichetti codes

Susanne Pumpluen

TL;DR

The paper introduces Menichetti algebras as nonassociative $G$-crossed product-like objects with a Galois extension $K/F$ in their nucleus, and extends the construction to generalized Menichetti algebras using central simple algebras. It develops division-algebra criteria via determinant formulas for left-multiplication maps, and applies these algebras as ambient spaces for linear codes by examining principal left ideals in their opposite algebras. Specific depth is given to cubic and quartic degree cases, including cyclic and biquadratic extensions, to produce concrete division algebras and illustrate rich nonassociative ambient structures for code design. Finally, the paper defines and analyzes linear Menichetti codes, including $(G,d)$-constacyclic codes, as left ideals in ambient nonassociative algebras, laying groundwork for broader code constructions over rings with noncyclic automorphism groups and pointing to future work on code distance and classification.

Abstract

We demonstrate the use of nonassociative algebras in code design and consider codes with nonassociative ambient algebras other than the well-known skew polycyclic codes. We define and investigate Menichetti algebras and identify them as important elements in the semiassociative Brauer monoid. Menichetti algebras can be viewed as generalisations of $G$-crossed product algebras; they are $n^2$-dimensional algebras with an $n$-dimensional Galois field extension $K/F$ with Galois group $G$ in their nucleus. We then extend the class of linear error-correcting codes obtained from left principal ideals in their ambient algebra using the opposite algebras of Menichetti algebras as ambient algebra. With the right choice of algebra they display symmetric and cyclic properties which promise efficient decoding algorithms. Well-known examples of such Menichetti codes are those skew constacyclic codes which have a nonassociative $G$-crossed product algebra (a nonassociative cyclic algebra) as their ambient algebra.

Menichetti's nonassociative $G$-crossed product algebras and Menichetti codes

TL;DR

The paper introduces Menichetti algebras as nonassociative -crossed product-like objects with a Galois extension in their nucleus, and extends the construction to generalized Menichetti algebras using central simple algebras. It develops division-algebra criteria via determinant formulas for left-multiplication maps, and applies these algebras as ambient spaces for linear codes by examining principal left ideals in their opposite algebras. Specific depth is given to cubic and quartic degree cases, including cyclic and biquadratic extensions, to produce concrete division algebras and illustrate rich nonassociative ambient structures for code design. Finally, the paper defines and analyzes linear Menichetti codes, including -constacyclic codes, as left ideals in ambient nonassociative algebras, laying groundwork for broader code constructions over rings with noncyclic automorphism groups and pointing to future work on code distance and classification.

Abstract

We demonstrate the use of nonassociative algebras in code design and consider codes with nonassociative ambient algebras other than the well-known skew polycyclic codes. We define and investigate Menichetti algebras and identify them as important elements in the semiassociative Brauer monoid. Menichetti algebras can be viewed as generalisations of -crossed product algebras; they are -dimensional algebras with an -dimensional Galois field extension with Galois group in their nucleus. We then extend the class of linear error-correcting codes obtained from left principal ideals in their ambient algebra using the opposite algebras of Menichetti algebras as ambient algebra. With the right choice of algebra they display symmetric and cyclic properties which promise efficient decoding algorithms. Well-known examples of such Menichetti codes are those skew constacyclic codes which have a nonassociative -crossed product algebra (a nonassociative cyclic algebra) as their ambient algebra.
Paper Structure (13 sections, 20 theorems, 114 equations)

This paper contains 13 sections, 20 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. The construction
  3. Preliminaries
  4. Menichetti algebras
  5. Classical associative crossed product algebras
  6. Generalized Menichetti algebras
  7. When a Menichetti algebra is a division algebra
  8. Cubic Field Extensions
  9. Menichetti and crossed product algebras of degree four
  10. Cyclic Field Extensions of Degree 4
  11. Menichetti algebras from biquadratic field extensions
  12. Nonassociative crossed product algebras from biquadratic field extensions
  13. Linear Menichetti codes

Key Result

Lemma 2.2

Define and identify $x_0z_0+\dots +x_{n-1}z_{n-1}$ with $(x_0,\dots,x_{n-1})$. Then the multiplication of the Menichetti algebra ${\rm Men}(K/F, k_0,\dots,k_{n-1})$ can also be expressed as

Theorems & Definitions (53)

  • Definition 2.1
  • Lemma 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Example 2.7
  • Remark 2.8
  • ...and 43 more