Self-Dual Cyclic Codes with Improved Minimum Distance Estimates via Extending the Chen-Ding Construction

Technical Summary: Self-Dual Cyclic Codes with Improved Minimum Distance Estimates via Extending the Chen-Ding Construction

Problem Statement
The construction of self-dual cyclic codes and the establishment of rigorous lower bounds on their minimum distances are fundamental challenges in coding theory. While Chen and Ding previously established a foundation for constructing self-dual cyclic codes in cases where the multiplicative order of $q$ modulo $n$, denoted $\text{ord}_n(q)$, is odd, the case where $\text{ord}_n(q)$ is even remained less explored regarding minimum distance properties. Furthermore, existing lower bounds for certain families of self-dual cyclic codes, particularly those derived from the Chen-Ding construction, were not always tight. The primary objective of this work is to extend the investigation to the case of even $\text{ord}_n(q)$, refine parameter selections for existing constructions, and derive tighter lower bounds on minimum distances, specifically investigating whether these codes can exceed square-root lower bounds.

Methodology
The paper employs algebraic coding theory techniques, focusing on the structural properties of cyclic codes over finite fields $\mathbb{F}_q$ (where $q$ is a power of 2). The methodology is built upon several key pillars:

1. Plotkin Sum Construction: The authors utilize the Plotkin sum method to construct self-dual cyclic codes of length $2n$ from a pair of codes $(C, C^\perp)$ or $(C, C^{\perp H})$ of length $n$. This approach transforms Euclidean or Hermitian dual-containing cyclic codes into self-dual codes.
2. Defining Set Analysis: A core analytical tool is the examination of the defining sets (sets of zeros) of the dual codes. The authors analyze the consecutive zero segments within these defining sets. By identifying specific patterns of consecutive zeros, they apply the BCH bound and its corollaries (such as the Roos bound) to estimate the minimum distance of the dual codes.
3. Parameter Refinement: The study introduces a refined selection of the designed distance $\delta$ for BCH codes. The authors demonstrate that reducing $\delta$ in the original code $C$ can paradoxically increase the minimum distance of its dual $C^\perp$ (or $C^{\perp H}$), thereby improving the overall minimum distance of the resulting self-dual code $C' = \text{Plotkin}(C, C^\perp)$.
4. Case Differentiation: The analysis is stratified based on the parity of $\text{ord}_n(q)$ (denoted as $m$) and the type of duality (Euclidean vs. Hermitian). Specific attention is given to cases where $m$ is even for Euclidean codes and $m$ is odd for Hermitian codes, which were previously less understood in terms of exact parameters.

Key Contributions and Results

• Extension to Even Order: The paper extends the Chen-Ding construction to the case where $\text{ord}_n(q)$ is even. It constructs Euclidean self-dual cyclic codes with even $m$ and Hermitian self-dual cyclic codes with even $m$, proving that their minimum distances satisfy square-root lower bounds.
• Improved Lower Bounds via Refined $\delta$: For Euclidean self-dual codes with odd $m$ and Hermitian self-dual codes with even $m$, the authors propose a new choice of designed distance $\delta$. This refinement leads to larger minimum distances for codes of the same length and dimension compared to previous constructions.
  • For Euclidean self-dual codes with $q=2^s$ ($s \ge 2$) and odd $m \ge 5$, the minimum distance is shown to be at least $q^{(m+1)/2} + 3$, improving upon the previous bound of $q^{(m+1)/2} - q + 1$.
  • For the binary case ($q=2$) with odd $m \ge 5$, the minimum distance is improved to at least $2^{(m+1)/2}$.
• Exact Parameter Determination: By leveraging known results on the defining sets of dual codes, the paper determines the exact parameters for two specific classes of self-dual cyclic codes:
  • Euclidean Self-Dual Codes (Even $m$): The code $C'$ constructed in Theorem 11 has exact parameters $[2(q^m - 1), q^m - 1, 2(q^{m/2} - 1)]_q$.
  • Hermitian Self-Dual Codes (Odd $m$): The code $C'$ constructed in Theorem 12 has exact parameters $[2(q^m - 1), q^m - 1, 2(q^{m_1} - 1)]_q$.
• Exceeding Square-Root Bounds: The authors establish that the Euclidean self-dual cyclic codes with even $\text{ord}_n(q)$ and Hermitian self-dual cyclic codes with odd $\text{ord}_n(q)$ have minimum distances that strictly exceed the square-root lower bound. This result addresses an open problem regarding the existence of infinite families of self-dual binary cyclic codes with minimum distances exceeding square-root bounds.

Significance and Claims
The paper claims to enrich the theory of self-dual cyclic codes by providing a more comprehensive framework for their construction and parameter estimation. Specifically:

• It resolves the construction and parameter determination for self-dual cyclic codes in cases involving even multiplicative orders, which were previously less characterized.
• It demonstrates that careful selection of the designed distance $\delta$ can yield tighter lower bounds on minimum distances, challenging the assumption that standard BCH bounds are always optimal for self-dual constructions.
• By determining exact parameters for specific families, the work confirms that these codes can surpass square-root lower bounds, a significant theoretical milestone.
• The authors modestly note that while their lower bounds are improved, there remains a discrepancy between these theoretical bounds and the actual minimum distances in practice. They identify the existence of tighter bounds that could exploit more zeros in the defining set (beyond the Roos bound) as an open problem for future investigation.

The work does not propose new experimental applications but rather provides theoretical advancements in the algebraic structure and distance properties of these codes, offering new insights for the estimation of lower bounds in coding theory.