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Quantum Bicyclic Hyperbolic Codes

Sankara Sai Chaithanya Rayudu, Pradeep Kiran Sarvepalli

TL;DR

This work addresses constructing quantum codes from 2D bicyclic hyperbolic codes by deriving explicit dual-containing criteria. Using the defining-set and cyclotomic coset framework, it provides a Euclidean dual-containing condition for primitive narrow-sense codes of length $n^2$ with a design-distance bound $d\le\delta$, and a Hermitian dual-containing condition for codes over $\mathbb{F}_{q^2}$ with $d\le\delta_h$, including non-primitive extensions. It demonstrates two quantum code constructions via CSS and Hermitian methods and highlights the role of coset structure in enabling dual containment. The results broaden the catalogue of quantum stabilizer codes derived from higher-dimensional cyclic codes and offer design guidelines for guaranteed minimum distance in the quantum setting.

Abstract

Bicyclic codes are a generalization of the one dimensional (1D) cyclic codes to two dimensions (2D). Similar to the 1D case, in some cases, 2D cyclic codes can also be constructed to guarantee a specified minimum distance. Many aspects of these codes are yet unexplored. Motivated by the problem of constructing quantum codes, in this paper, we study some structural properties of certain bicyclic codes. We show that a primitive narrow-sense bicyclic hyperbolic code of length $n^2$ contains its dual if and only if its design distance is lower than $n-Δ$, where $Δ=\mathcal{O}(\sqrt{n})$. We extend the sufficiency condition to the non-primitive case as well. We also show that over quadratic extension fields, a primitive bicyclic hyperbolic code of length $n^2$ contains Hermitian dual if and only if its design distance is lower than $n-Δ_h$, where $Δ_h=\mathcal{O}(\sqrt{n})$. Our results are analogous to some structural results known for BCH and Reed-Solomon codes. They further our understanding of bicyclic codes. We also give an application of these results by showing that we can construct two classes of quantum bicyclic codes based on our results.

Quantum Bicyclic Hyperbolic Codes

TL;DR

This work addresses constructing quantum codes from 2D bicyclic hyperbolic codes by deriving explicit dual-containing criteria. Using the defining-set and cyclotomic coset framework, it provides a Euclidean dual-containing condition for primitive narrow-sense codes of length with a design-distance bound , and a Hermitian dual-containing condition for codes over with , including non-primitive extensions. It demonstrates two quantum code constructions via CSS and Hermitian methods and highlights the role of coset structure in enabling dual containment. The results broaden the catalogue of quantum stabilizer codes derived from higher-dimensional cyclic codes and offer design guidelines for guaranteed minimum distance in the quantum setting.

Abstract

Bicyclic codes are a generalization of the one dimensional (1D) cyclic codes to two dimensions (2D). Similar to the 1D case, in some cases, 2D cyclic codes can also be constructed to guarantee a specified minimum distance. Many aspects of these codes are yet unexplored. Motivated by the problem of constructing quantum codes, in this paper, we study some structural properties of certain bicyclic codes. We show that a primitive narrow-sense bicyclic hyperbolic code of length contains its dual if and only if its design distance is lower than , where . We extend the sufficiency condition to the non-primitive case as well. We also show that over quadratic extension fields, a primitive bicyclic hyperbolic code of length contains Hermitian dual if and only if its design distance is lower than , where . Our results are analogous to some structural results known for BCH and Reed-Solomon codes. They further our understanding of bicyclic codes. We also give an application of these results by showing that we can construct two classes of quantum bicyclic codes based on our results.

Paper Structure

This paper contains 7 sections, 13 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Bicyclic codes
  4. Bicyclic hyperbolic codes
  5. Euclidean Dual Containing Bicyclic Codes
  6. Hermitian Dual Containing Codes
  7. Conclusion

Key Result

Lemma 3

Let $C$ be the bicyclic code of length $n_1\times n_2$ over $\mathbb{F}_q$ such that $\text{gcd}(n_i,q) = 1$ and let $Z$ be the defining set of $C$. Then $C$ contains its Euclidean dual $C^\perp$ if and only if where $Z^{-1} = \left\{ (-x \bmod n_1,-y \bmod n_2)\ \middle|\ (x,y)\in Z\right\}$.

Theorems & Definitions (23)

  • Definition 1: Bicyclic codes
  • Definition 2: Bicyclic hyperbolic codes
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 5: Euclidean dual containing bicyclic codes
  • proof
  • Lemma 6
  • proof
  • ...and 13 more