[Paper Review] Quantum Block and Synchronizable Codes Derived from Certain Classes of Polynomials

Tao Zhang, Gennian Ge|arXiv (Cornell University)|Aug 5, 2015

Quantum Computing Algorithms and Architecture19 references3 citations

TL;DR

This paper presents a novel construction of quantum block codes and quantum synchronizable codes using polynomials over finite fields, specifically leveraging duadic codes over $\mathbb{F}_2$. By exploiting the algebraic structure of cyclotomic cosets and irreducible factors of certain polynomials, the authors derive new quantum codes with improved parameters, including an infinite family of quantum synchronizable codes with maximal misalignment tolerance, surpassing existing constructions in parameter range and efficiency.

ABSTRACT

One central theme in quantum error-correction is to construct quantum codes that have a large minimum distance. In this paper, we first present a construction of classical codes based on certain class of polynomials. Through these classical codes, we are able to obtain some new quantum codes. It turns out that some of quantum codes exhibited here have better parameters than the ones available in the literature. Meanwhile, we give a new class of quantum synchronizable codes with highest possible tolerance against misalignment from duadic codes.

Motivation & Objective

• To develop a new class of quantum block codes with enhanced minimum distance using polynomial-based classical codes.
• To address the scarcity of infinite families of quantum synchronizable codes with high misalignment tolerance.
• To generalize existing constructions of quantum synchronizable codes by exploiting the algebraic structure of duadic codes over $\mathbb{F}_2$.
• To provide explicit constructions of quantum codes with better parameters than those listed in current online tables.

Proposed method

• Construct classical linear codes via evaluation of polynomials over finite fields, particularly focusing on irreducible factors of $d_{n0}(x)$ over $\mathbb{F}_2$.
• Utilize the $2$-cyclotomic cosets modulo $p^n$ to decompose the defining sets of cyclic codes, where $p \equiv -1 \pmod{8}$ is an odd prime.
• Apply the generator polynomials $f_1(x)$ and $f_2(x)$, formed as products of irreducible factors $h_{ij}(x)$, to define cyclic codes $D_1$ and $D_2$.
• Leverage the self-orthogonality and duality properties of these codes to construct quantum codes via the CSS construction.
• Construct quantum synchronizable codes by extending the code length with $a_l$ and $a_r$ padding, preserving the required symmetry and algebraic structure.
• Use the order of 2 modulo $p^n$, denoted $\text{ord}_{p^n}(2) = t_n$, to determine the degree and number of irreducible factors, enabling precise control over code parameters.

Experimental results

Research questions

• RQ1 Can polynomial-based constructions yield quantum codes with better parameters than existing ones?
• RQ2 What algebraic structures enable the construction of quantum synchronizable codes with maximal tolerance to misalignment?
• RQ3 How can duadic codes over $\mathbb{F}_2$ be systematically used to generate infinite families of quantum synchronizable codes?
• RQ4 What role do cyclotomic cosets and irreducible factorizations of $d_{n0}(x)$ play in determining code parameters?

Key findings

• The authors construct a new infinite family of quantum synchronizable codes with parameters $(a_l, a_r)$-$[[p^n + a_l + a_r, k]]_2$ where $a_l + a_r < p^n$, achieving maximal misalignment tolerance.
• For $p = 31$, $n = 2$, the construction yields quantum synchronizable codes such as $(a_l, a_r)$-$[[961 + a_l + a_r, 621]]_2$ with $k = 621$ when $f_1(x) = \prod_{i=1}^3 h_{1i}(x) \prod_{j=1}^3 h_{2j}(x)$ and $f_2(x) = \prod_{i=1}^3 h_{1i}(x)$.
• The construction generalizes prior results, including those in [26], by allowing a broader range of code lengths and parameters through flexible choices of $u_i$ and $v_i$.
• The method produces quantum codes with improved parameters compared to those listed in online tables [6] and [12], particularly in terms of minimum distance and rate.
• The number of irreducible factors of $d_{n0}(x)$ over $\mathbb{F}_2$ is $\frac{p^{n-1}(p-1)}{2t_n}$, where $t_n = \text{ord}_{p^n}(2)$, enabling precise control over code dimension and length.
• Theoretical analysis confirms that the codes are self-orthogonal and satisfy the necessary conditions for quantum code construction via the CSS method, ensuring error correction capability.