[Paper Review] Construction of Z4-linear Reed-Muller codes

Jaume Pujol, Josep Rifà+1 more|ArXiv.org|Jan 19, 2008

Coding theory and cryptography11 references4 citations

TL;DR

This paper introduces a new family of ${\mathbb{Z}}_4$-linear Reed-Muller codes constructed via quaternary Plotkin-type constructions, ensuring that under the Gray map, the resulting binary codes match the parameters and structural properties (including duality and inclusion) of classical binary linear Reed-Muller codes. The key contribution is a systematic construction where the first-order and $(m-2)$-th order codes map to ${\mathbb{Z}}_4$-linear Hadamard and extended perfect codes, respectively.

ABSTRACT

New quaternary Plotkin constructions are given and are used to obtain new families of quaternary codes. The parameters of the obtained codes, such as the length, the dimension and the minimum distance are studied. Using these constructions new families of quaternary Reed-Muller codes are built with the peculiarity that after using the Gray map the obtained Z4-linear codes have the same parameters and fundamental properties as the codes in the usual binary linear Reed-Muller family. To make more evident the duality relationships in the constructed families the concept of Kronecker inner product is introduced.

Motivation & Objective

• To construct new families of ${\mathbb{Z}}_4$-linear codes whose Gray map images replicate the parameters and fundamental properties of classical binary linear Reed-Muller (RM) codes.
• To generalize the binary Plotkin construction to the quaternary domain for ${\mathbb{Z}}_4$-linear codes, enabling systematic generation of RM-like families.
• To establish duality relationships in the ${\mathbb{Z}}_4$-linear setting using the Kronecker inner product, mirroring the duality of binary RM codes.
• To ensure that the first-order and $(m-2)$-th order codes in the new family map to ${\mathbb{Z}}_4$-linear Hadamard and extended 1-perfect codes, respectively, under the Gray map.

Proposed method

• The authors introduce a quaternary Plotkin construction based on Kronecker products of generating matrices, generalizing the binary Plotkin method to ${\mathbb{Z}}_4$.
• They define a new family of ${\mathbb{Z}}_4$-linear codes, denoted ${\mathcal{RM}}_s(r,m)$, for $0 \leq r \leq m$ and $0 \leq s \leq \lfloor (m-1)/2 \rfloor$, using recursive matrix constructions.
• The Kronecker inner product is introduced to define duality in the ${\mathbb{Z}}_4$-linear setting, ensuring ${\mathcal{RM}}_s(r,m)$ and ${\mathcal{RM}}_s(m-1-r,m)$ are dual codes.
• The Gray map $\Phi$ is applied to transform ${\mathbb{Z}}_4$-linear codes into binary codes, preserving length, dimension, and minimum distance.
• Monomial equivalence is used to construct an alternative code family $\overline{{\mathcal{RM}}}_s(r,m)$ such that it is dual to ${\mathcal{RM}}_s(m-1-r,m)$ under the standard inner product.
• The construction is verified inductively, proving that the resulting codes satisfy the same inclusion, duality, and parameter properties as classical binary RM codes.

Experimental results

Research questions

• RQ1 Can a quaternary generalization of the binary Plotkin construction yield ${\mathbb{Z}}_4$-linear codes whose Gray map images match the parameters of classical binary linear Reed-Muller codes?
• RQ2 Do the first-order and $(m-2)$-th order codes in the new family map to ${\mathbb{Z}}_4$-linear Hadamard and extended 1-perfect codes under the Gray map?
• RQ3 Can a duality relationship in the ${\mathbb{Z}}_4$-linear setting be defined such that it mirrors the duality of binary RM codes?
• RQ4 Is it possible to construct a family of ${\mathbb{Z}}_4$-linear codes with the same inclusion, dimension, and minimum distance progression as binary RM codes?

Key findings

• The constructed ${\mathbb{Z}}_4$-linear Reed-Muller codes ${\mathcal{RM}}_s(r,m)$ have length $n = 2^m$, dimension $k = \sum_{i=0}^r \binom{m}{i}$, and minimum distance $d = 2^{m-r}$, matching the parameters of classical binary RM codes.
• The first-order code ${\mathcal{RM}}_s(1,m)$ maps under the Gray map to a ${\mathbb{Z}}_4$-linear Hadamard code, and the $(m-2)$-th order code ${\mathcal{RM}}_s(m-2,m)$ maps to a ${\mathbb{Z}}_4$-linear extended 1-perfect code.
• The codes ${\mathcal{RM}}_s(r,m)$ and ${\mathcal{RM}}_s(m-1-r,m)$ are duals under the Kronecker inner product, replicating the duality of binary RM codes.
• The code ${\mathcal{RM}}_s(0,m)$ is the repetition code generated by the all-2s vector, and ${\mathcal{RM}}_s(m,m)$ is the full space ${\mathbb{Z}}_4^{2^{m-1}}$.
• The code ${\mathcal{RM}}_s(r-1,m)$ is a subcode of ${\mathcal{RM}}_s(r,m)$ for $r > 0$, preserving the inclusion hierarchy of binary RM codes.
• An alternative code family $\overline{{\mathcal{RM}}}_s(r,m)$, monomially equivalent to ${\mathcal{RM}}_s(r,m)$, is constructed such that it is dual to ${\mathcal{RM}}_s(m-1-r,m)$ under the standard inner product.