Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Special matrices over finite fields and their applications to quantum error-correcting codes

Meng Cao

TL;DR

This work extends matrix-product codes by analyzing NSC-defining matrices $A$ with $AA^{\dag}$ $(D,\tau)$-monomial to enhance quantum code design. It derives a closed-form formula for the Hermitian hull dimension and furnishes necessary/sufficient conditions for MP codes to be Hermitian dual-containing (HDC), almost-Hermitian dual-containing (AHDC), Hermitian self-orthogonal (HSO), almost Hermitian self-orthogonal (AHSO), and Hermitian LCD. It further characterizes the τ≠$\mathbbm{1}_{k}$ case, providing explicit criteria and alternative AHDC/AHSO formulations, and presents both trivial and nontrivial construction methods for HDC/AHDC MP codes, including distance-lower-bound considerations. The results enable systematic design of MP-based quantum codes with desired dual properties and distance guarantees, and they quantify how constituent-code relations influence achievable code properties. Overall, the paper offers practical criteria and constructions to build robust quantum error-correcting codes via structured MP codes.

Abstract

The matrix-product (MP) code $\mathcal{C}_{A,k}:=[\mathcal{C}_{1},\mathcal{C}_{2},\ldots,\mathcal{C}_{k}]\cdot A$ with a non-singular by column (NSC) matrix $A$ plays an important role in constructing good quantum error-correcting codes. In this paper, we study the MP code when the defining matrix $A$ satisfies the condition that $AA^†$ is $(D,τ)$-monomial. We give an explicit formula for calculating the dimension of the Hermitian hull of a MP code. We provide the necessary and sufficient conditions that a MP code is Hermitian dual-containing (HDC), almost Hermitian dual-containing (AHDC), Hermitian self-orthogonal (HSO), almost Hermitian self-orthogonal (AHSO), and Hermitian LCD, respectively. We theoretically determine the number of all possible ways involving the relationships among the constituent codes to yield a MP code with these properties, respectively. We give alternative necessary and sufficient conditions for a MP code to be AHDC and AHSO, respectively, and show several cases where a MP code is not AHDC or AHSO. We provide the construction methods of HDC and AHDC MP codes, including those with optimal minimum distance lower bounds.

Special matrices over finite fields and their applications to quantum error-correcting codes

TL;DR

This work extends matrix-product codes by analyzing NSC-defining matrices with -monomial to enhance quantum code design. It derives a closed-form formula for the Hermitian hull dimension and furnishes necessary/sufficient conditions for MP codes to be Hermitian dual-containing (HDC), almost-Hermitian dual-containing (AHDC), Hermitian self-orthogonal (HSO), almost Hermitian self-orthogonal (AHSO), and Hermitian LCD. It further characterizes the τ≠ case, providing explicit criteria and alternative AHDC/AHSO formulations, and presents both trivial and nontrivial construction methods for HDC/AHDC MP codes, including distance-lower-bound considerations. The results enable systematic design of MP-based quantum codes with desired dual properties and distance guarantees, and they quantify how constituent-code relations influence achievable code properties. Overall, the paper offers practical criteria and constructions to build robust quantum error-correcting codes via structured MP codes.

Abstract

The matrix-product (MP) code with a non-singular by column (NSC) matrix plays an important role in constructing good quantum error-correcting codes. In this paper, we study the MP code when the defining matrix satisfies the condition that is -monomial. We give an explicit formula for calculating the dimension of the Hermitian hull of a MP code. We provide the necessary and sufficient conditions that a MP code is Hermitian dual-containing (HDC), almost Hermitian dual-containing (AHDC), Hermitian self-orthogonal (HSO), almost Hermitian self-orthogonal (AHSO), and Hermitian LCD, respectively. We theoretically determine the number of all possible ways involving the relationships among the constituent codes to yield a MP code with these properties, respectively. We give alternative necessary and sufficient conditions for a MP code to be AHDC and AHSO, respectively, and show several cases where a MP code is not AHDC or AHSO. We provide the construction methods of HDC and AHDC MP codes, including those with optimal minimum distance lower bounds.
Paper Structure (8 sections, 14 theorems, 20 equations, 1 table)

This paper contains 8 sections, 14 theorems, 20 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Several types of MP codes
  4. Further characterization of Theorem 3.4 for $\tau\neq\mathbbm{1}_{k}$
  5. An explicit characterization of Theorem 3.4 for $\tau\neq\mathbbm{1}_{k}$
  6. Another characterization of AHDC and AHSO MP codes
  7. HDC MP codes
  8. Conclusion

Key Result

Lemma 3.1

( Guenda2020Linear) Let $\mathcal{C}_{i}$ be an $[n,t_{i},d_{i}]_{q^{2}}$ code with a generator matrix $G_{i}$ for $i=1,2$. Then $\mathrm{dim}(\mathcal{C}_{1}\cap\mathcal{C}_{2}^{\bot_{H}})=t_{1}-\mathrm{rank}(G_{1}G_{2}^{\dag})$.

Theorems & Definitions (23)

  • Definition 1.1
  • Lemma 3.1
  • Corollary 3.2
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Corollary 3.5
  • Lemma 4.1
  • proof
  • ...and 13 more