[Paper Review] Families of Hadamard Z2Z4Q8-codes

Ángel del Rı́o, Josep Rifà|arXiv (Cornell University)|Nov 22, 2012

Coding theory and cryptography7 references3 citations

TL;DR

This paper introduces and characterizes a new class of Hadamard codes over the group ring ℤ₂ℤ₄𝒬₈, demonstrating the existence of 'pure' ℤ₂ℤ₄𝒬₈-codes that are not equivalent to any ℤ₂ℤ₄-linear codes. It establishes bounds on the rank and kernel dimension for such codes, provides constructions via Gray maps and Kronecker products, and identifies specific examples—including a non-ℤ₂ℤ₄-linear Hadamard code of length 32 and a shape-4 code—showcasing the richness of this code family beyond traditional linear models.

ABSTRACT

A Z2Z4Q8-code is a non-empty subgroup of a direct product of copies of Z_2, Z_4 and Q_8 (the binary field, the ring of integers modulo 4 and the quaternion group on eight elements, respectively). Such Z2Z4Q8-codes are translation invariant propelinear codes as the well known Z_4-linear or Z_2Z_4-linear codes. In the current paper, we show that there exist "pure" Z2Z4Q8-codes, that is, codes that do not admit any abelian translation invariant propelinear structure. We study the dimension of the kernel and rank of the Z2Z4Q8-codes, and we give upper and lower bounds for these parameters. We give tools to construct a new class of Hadamard codes formed by several families of Z2Z4Q8-codes; we study and show the different shapes of such a codes and we improve the upper and lower bounds for the rank and the dimension of the kernel when the codes are Hadamard.

Motivation & Objective

• To investigate the structural properties of ℤ₂ℤ₄𝒬₈-codes, a generalization of ℤ₂ℤ₄-linear codes using the quaternion group 𝒬₈.
• To determine whether such codes can exist that are not equivalent to any ℤ₂ℤ₄-linear code, thus establishing the existence of 'pure' ℤ₂ℤ₄𝒬₈-codes.
• To analyze the rank and dimension of the kernel for these codes, especially in the context of Hadamard codes.
• To construct infinite families of Hadamard ℤ₂ℤ₄𝒬₈-codes using Gray maps and Kronecker products, extending known constructions.
• To classify the possible group structures and shapes of Hadamard ℤ₂ℤ₄𝒬₈-codes and improve bounds on their invariants.

Proposed method

• The paper defines ℤ₂ℤ₄𝒬₈-codes as subgroups of ℤ₂ᵏ¹ × ℤ₄ᵏ² × 𝒬₈ᵏ³, leveraging the group structure to define translation-invariant propelinear codes.
• It employs the Gray map Φ to transform ℤ₂ℤ₄𝒬₈-codes into binary codes, preserving Hamming distance and enabling the study of their binary image properties.
• The Kronecker product construction 𝒦_g is used to generate new codes from existing ones, particularly to increase code length and explore new shapes.
• Group-theoretic tools are applied to analyze the structure of the code group, including the use of swappers to detect new elements in the group generated by generators.
• The paper uses the concept of 'shape' to classify the group structure of ℤ₂ℤ₄𝒬₈-codes, with five possible shapes identified.
• Specific examples are constructed using generators over ℤ₂, ℤ₄, and 𝒬₈, and verified via the Gray map to confirm their binary image is a Hadamard code with desired parameters.

Experimental results

Research questions

• RQ1 Do there exist ℤ₂ℤ₄𝒬₈-codes that are not equivalent to any ℤ₂ℤ₄-linear code, i.e., 'pure' ℤ₂ℤ₄𝒬₈-codes?
• RQ2 What are the tight upper and lower bounds for the rank and dimension of the kernel in ℤ₂ℤ₄𝒬₈-codes, especially for Hadamard codes?
• RQ3 Can all possible shapes of ℤ₂ℤ₄𝒬₈-codes be realized, and if so, how can they be constructed?
• RQ4 Are there Hadamard codes of length 16 or 32 that are not ℤ₂ℤ₄-linear but can be realized as ℤ₂ℤ₄𝒬₈-codes?
• RQ5 What is the maximum possible rank for a Hadamard ℤ₂ℤ₄𝒬₈-code of a given shape, and can such codes be constructed explicitly?

Key findings

• The paper constructs the first known example of a 'pure' ℤ₂ℤ₄𝒬₈-code that is not equivalent to any ℤ₂ℤ₄-linear code, proving such codes exist.
• A Hadamard code of length 32 with type (3,0,3), shape 3, rank 7, and kernel dimension 4 is constructed as a ℤ₂ℤ₄𝒬₈-code, which is not ℤ₂ℤ₄-linear.
• A new Hadamard code of length 64 with type (3,0,4), shape 2, rank 8, and kernel dimension 3 is constructed via the Kronecker product of a base code.
• The paper provides a complete classification of possible group structures for Hadamard ℤ₂ℤ₄𝒬₈-codes, identifying five distinct shapes, with explicit constructions for shapes 1, 2, 3, 4, and 5.
• A shape-4 code is explicitly constructed with generators over ℤ₂⁴ × 𝒬₈, and a variation of the Kronecker construction yields a non-linear code of rank 6, demonstrating the maximum rank for this shape.
• The paper improves the upper and lower bounds for the rank and kernel dimension of Hadamard ℤ₂ℤ₄𝒬₈-codes, showing they are tighter than those for general ℤ₂ℤ₄𝒬₈-codes.