[Paper Review] A group induced four-circulant construction for self-dual codes and new extremal binary self-dual codes

Joe Gildea, Abidin Kaya+2 more|arXiv (Cornell University)|Dec 26, 2019

Coding theory and cryptography19 references4 citations

TL;DR

This paper introduces a novel group-induced four-circulant construction over group rings to generate extremal and optimal binary self-dual codes. By leveraging group rings over $φ_2$, $φ_2 + uφ_2$, and $φ_4 + uφ_4$ with groups of order 3, 7, 9, 13, and 15, the authors construct 5 new $[56,28,10]$ codes, 23 new extremal $[68,34,14]$ codes with distinct weight enumerators, and 15 new $[80,40,14]$ codes, significantly expanding the known set of extremal self-dual codes.

ABSTRACT

We introduce an altered version of the four circulant construction over group rings for self-dual codes. We consider this construction over the binary field, the rings F_2 + uF_2 and F_4 + uF_4; using groups of order 3, 7, 9, 13, and 15. Through these constructions and their extensions, we find binary self-dual codes of lengths 32, 40, 56, 64, 68 and 80, all of which are extremal or optimal. In particular, we find five new self-dual codes of parameters [56, 28, 10], twenty-three extremal binary self-dual codes of length 68 with new weight enumerators and fifteen new self-dual codes of parameters [80, 40, 14].

Motivation & Objective

• To develop a new algebraic construction method for self-dual codes using group rings and modified four-circulant matrices.
• To extend existing four-circulant constructions by replacing circulant matrices with group ring-generated matrices over rings of characteristic 2.
• To discover new extremal and optimal binary self-dual codes of lengths 32, 40, 56, 64, 68, and 80.
• To establish a theoretical link between units/non-units in group rings and the self-duality of the resulting codes.
• To demonstrate the effectiveness of the construction through computational enumeration using MAGMA.

Proposed method

• The construction uses a generator matrix of the form $[I_{2n} \mid \begin{smallmatrix} A & B \\ B^T & A^T \end{smallmatrix}]$, where $A$ and $B$ are matrices derived from group ring elements over $\mathbb{F}_2$, $\mathbb{F}_2 + u\mathbb{F}_2$, and $\mathbb{F}_4 + u\mathbb{F}_4$.
• The method applies group rings of order 3, 7, 9, 13, and 15 to generate structured matrices that are not necessarily circulant, enabling new code constructions.
• Self-duality is ensured by verifying that the matrix satisfies the condition $AA^T + BB^T = -I_n$ over rings of characteristic 2, where signs are adjusted accordingly.
• The construction is extended via the neighboring method to generate additional self-dual codes from existing ones, particularly for length 68.
• Computational verification is performed using the MAGMA algebra system to generate and test the codes for self-duality and extremality.
• Weight enumerators are computed and compared to known lists to confirm novelty, especially for codes of length 68 and 80.

Experimental results

Research questions

• RQ1 Can a modified four-circulant construction over group rings produce new extremal binary self-dual codes?
• RQ2 What is the role of units and non-units in group rings in generating self-dual codes?
• RQ3 How do different group rings and group orders influence the structure and parameters of the resulting self-dual codes?
• RQ4 Can the neighboring construction method yield new extremal codes from existing ones in the new family?
• RQ5 What is the extent of novelty in the weight enumerators of the constructed codes, particularly for lengths 68 and 80?

Key findings

• The authors constructed five new extremal binary self-dual codes of parameters $[56,28,10]$ with distinct weight enumerators in $W_{56,1}$.
• Twenty-three new extremal binary self-dual codes of length 68 were found, with weight enumerators in $W_{68,2}$ and parameters $(\gamma=3, \beta \in \{165,169,171,173\})$, $(\gamma=4, \beta \in \{163,165,173,177,179,181,183,185,187,188,189,190,192,193,204,208,210,214\})$, and $(\gamma=5, \beta=201)$.
• Fifteen new extremal binary self-dual codes of length 80 were constructed with parameters $[80,40,14]$, having weight enumerators in $W_{80,2}$ with $\beta=1$ and $\alpha \in \{-96,-150,-168,-186,-204,-222,-240,-258,-312\}$, and $\beta=10$ with $\alpha \in \{-204,-276,-294,-330,-348,-366\}$.
• The construction successfully generated new codes across multiple lengths—32, 40, 56, 64, 68, and 80—many of which are extremal or optimal.
• The neighboring construction method yielded 17 additional new codes of length 68, all with automorphism group of order 2 and distinct weight enumerators.
• The study confirms that group ring-based constructions, especially with non-circulant matrices, are a powerful and fertile source for discovering new extremal self-dual codes.