[Paper Review] Notes on Reed-Muller Codes

Yanling Chen, A. J. Han Vinck|ArXiv.org|Jan 14, 2009

Coding theory and cryptography4 references3 citations

TL;DR

This paper establishes the uniqueness of the first-order Reed-Muller (RM) code among all linear codes with the same length, dimension, and minimum distance. It further constructs a new family of linear subcodes for the second-order RM code when the length is a power of two with even exponent, demonstrating that these subcodes achieve the best-known or upper-bound minimum distances for their parameters. An additional contribution is the discovery of a commutative additive group of full-rank symplectic matrices derived from these codes.

ABSTRACT

In this paper, we consider the Reed-Muller (RM) codes. For the first order RM code, we prove that it is unique in the sense that any linear code with the same length, dimension and minimum distance must be the first order RM code; For the second order RM code, we give a constructive linear sub-code family for the case when m is even. This is an extension of Corollary 17 of Ch. 15 in the coding book by MacWilliams and Sloane. Furthermore, we show that the specified sub-codes of length <= 256 have minimum distance equal to the upper bound or the best known lower bound for all linear codes of the same length and dimension. As another interesting result, we derive an additive commutative group of the symplectic matrices with full rank.

Motivation & Objective

• To prove that any linear code with parameters $[2^m, 1+m, 2^{m-1}]$ must be equivalent to the first-order Reed-Muller code.
• To extend the known subcode families of the second-order RM code $\mathcal{R}(2,m)$ to the case where $m$ is even, which had previously only been established for odd $m$.
• To demonstrate that the constructed subcodes for $m \leq 8$ achieve the upper bound or best-known minimum distance for their length and dimension.
• To reveal a new algebraic structure: an additive commutative group of $m \times m$ symplectic matrices with full rank, derived from the code's coset structure.

Proposed method

• Proves uniqueness of the first-order RM code by analyzing the dual code of a truncated generator matrix and applying the Plotkin bound to eliminate lower minimum distance cases.
• Constructs a linear subcode family for $\mathcal{R}(2,m)$ when $m$ is even using a systematic extension of cyclic subcodes with specific idempotents.
• Employs the Grey-Rankin bound to show that a self-complementary linear code with parameters $[2^m, 1+3m/2, 2^{m-1}-2^{m/2-1}]$ cannot have a larger minimum distance.
• Derives the symplectic matrix group by associating each coset of $\mathcal{R}(1,m)$ in the subcode with a quadratic bent function, then forming a matrix set closed under addition.
• Uses the fact that the subcode is linear to show that the set of associated symplectic matrices, including the zero matrix, forms a commutative additive group.
• Verifies the constructed subcodes' minimum distances by comparing them to known upper bounds and best-known values from the literature (e.g., [1]).

Experimental results

Research questions

• RQ1 Is the first-order Reed-Muller code the only linear code with parameters $[2^m, 1+m, 2^{m-1}]$ up to equivalence?
• RQ2 Can a linear subcode family of $\mathcal{R}(2,m)$ be constructed for even $m$, extending the known result for odd $m$?
• RQ3 Do the constructed subcodes for $m \leq 8$ achieve the theoretical upper bound or the best-known minimum distance for their length and dimension?
• RQ4 What algebraic structure underlies the coset representatives of the constructed subcodes, particularly in relation to symplectic matrices?

Key findings

• The first-order Reed-Muller code is uniquely characterized by its parameters $[2^m, 1+m, 2^{m-1}]$, meaning any linear code with these parameters is equivalent to it.
• A new linear subcode family of $\mathcal{R}(2,m)$ is constructed for even $m$, extending Corollary 17 of Chapter 15 in MacWilliams and Sloane's book.
• For $m \leq 8$, all constructed subcodes achieve either the theoretical upper bound $d_+$ or the best-known lower bound $d_-$ for minimum distance among linear codes of the same length and dimension.
• The subcodes contain $\mathcal{R}(1,m)$ and have weight distributions that match the specified subcode families, confirming their optimality.
• An additive commutative group $\mathcal{G}$ of $m \times m$ symplectic matrices with $2^{m/2}$ elements is constructed, where all non-zero matrices have full rank.
• This group $\mathcal{G}$ arises naturally from the coset structure of the subcode $\mathcal{R}_{2t+2}^{t+1}$, and is closed under matrix addition.