Binary reed-muller codes show dense range of weights from 336 up
Some Results on the Weight Spectrum of Binary Reed-Muller Codes $RM(m-7,m)$
Information Theory
Summary
This paper studies certain mathematical objects called binary Reed-Muller codes, which are used in error correction. The authors found that for these codes with specific parameters, all even weight values between 336 and a large upper bound occur, along with some identified low weights. They proved this by starting with a smaller base case and then extending it to larger cases using a technique called concatenation. They also examined some remaining unexplained weights and showed that several common methods cannot produce them, helping to narrow down the open problems.
What this means in practice
- •For error correction engineers: Design error-correcting codes with predictable weight distributions for better communication reliability in systems using Reed-Muller codes with high order parameters.
- •For cryptographic hardware designers: Assess code weight properties to analyze and optimize hardware implementations reliant on Reed-Muller codes for secure data transmission.
Authors
Zibo Wei, Bo Zhou, Xiao Zhang, Hao Chen
Abstract
We study the set of weights occurring in the high-order binary Reed-Muller codes $RM(m-7,m)$. For every $m\geq 14$, we prove that the weight spectrum contains all even weights in $[336,2^m-336]$, together with an explicit finite set of low weights and their complements. The proof first establishes the corresponding base-code inclusion in $RM(7,14)$. This base step uses four-block concatenation, explicit degree-six monomial witnesses, and known spectrum inputs for $RM(5,10)$ and $RM(6,12)$. The base inclusion is then lifted to all $m\geq 14$ by a concatenation-based induction. We also analyze the remaining base-code target weights $322,326,330,334$ in several restricted families, including four-block branches, affine-flat and structured inner corrections in a fixed Carlet-type Maiorana-McFarland outer shell, and sparse sums of degree-7 coordinate monomials. The residual analysis delineates several construction routes that cannot yield the four unresolved base-code weights, thereby sharpening the remaining $c=7$ spectrum problem.