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Some New Non-binary Quantum Codes from One-generator Quasi-cyclic Codes

Tushar Bag, Hai Q Dinh, Daniel Panario

TL;DR

This work develops general necessary and sufficient conditions for symplectic self-orthogonality and symplectic dual-containing properties of one-generator and two-generator quasi-cyclic codes over finite fields, using both matrix and polynomial formalisms. By enforcing these conditions, the authors construct quantum error-correcting codes with record-breaking parameters, demonstrated through multiple examples across different primes and lengths, notably yielding $[[11,0,6]]$, $[[13,6,4]]$, $[[23,12,5]]$, and $[[16,6,5]]$ codes. The results bridge QC-code structure with QECC design, and the polynomial approach via transpose and parity-check formulations provides practical criteria for code construction. The paper also discusses limitations in dual-containing constructions for two-generator QC codes and suggests future directions, including skew quasi-cyclic codes, to uncover further high-performance quantum codes.

Abstract

This article studies one-generator and two-generator quasi-cyclic codes over finite fields. We present two versions of necessary and sufficient conditions for the symplectic selforthogonality of one-generator quasi-cyclic codes, using both matrix and polynomial approaches. We provide two versions of necessary and sufficient conditions for two-generator quasi-cyclic codes for symplectic self-orthogonality and the symplectic dual-containing condition. Additionally, using these necessary and sufficient conditions, we construct new quantum codes with record-breaking parameters that improve upon current records.

Some New Non-binary Quantum Codes from One-generator Quasi-cyclic Codes

TL;DR

This work develops general necessary and sufficient conditions for symplectic self-orthogonality and symplectic dual-containing properties of one-generator and two-generator quasi-cyclic codes over finite fields, using both matrix and polynomial formalisms. By enforcing these conditions, the authors construct quantum error-correcting codes with record-breaking parameters, demonstrated through multiple examples across different primes and lengths, notably yielding , , , and codes. The results bridge QC-code structure with QECC design, and the polynomial approach via transpose and parity-check formulations provides practical criteria for code construction. The paper also discusses limitations in dual-containing constructions for two-generator QC codes and suggests future directions, including skew quasi-cyclic codes, to uncover further high-performance quantum codes.

Abstract

This article studies one-generator and two-generator quasi-cyclic codes over finite fields. We present two versions of necessary and sufficient conditions for the symplectic selforthogonality of one-generator quasi-cyclic codes, using both matrix and polynomial approaches. We provide two versions of necessary and sufficient conditions for two-generator quasi-cyclic codes for symplectic self-orthogonality and the symplectic dual-containing condition. Additionally, using these necessary and sufficient conditions, we construct new quantum codes with record-breaking parameters that improve upon current records.

Paper Structure

This paper contains 8 sections, 15 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. One-generator QC codes
  4. Two-generators QC codes
  5. Self-orthogonal QC codes
  6. Dual-containing QC codes
  7. QECCs from QC codes
  8. Conclusion and Future work

Key Result

Theorem 3.1

Let $C$ be a one-generator QC code of length $2n$ and index $2$ over $\mathbb F_q$. Then a generator of $C$ is of the form $(r_1(x)g_1(x), r_2(x)g_2(x))$, where ${g_i}(x)h_i(x)= x^n -1$ and $\gcd(r_i(x),h_i(x))=1$ for $i = 1, 2$.

Theorems & Definitions (44)

  • Definition 2.1
  • Theorem 3.1
  • proof
  • Definition 3.2
  • Theorem 3.3
  • proof
  • Remark 3.4
  • Example 3.5
  • Corollary 3.6
  • Theorem 3.7
  • ...and 34 more