Small and Large Weights in Steinhaus Triangles

TL;DR

This paper investigates the distribution of weights in binary Steinhaus triangles, identifying specific weight values and their generating sequences, and providing partial insights into the second-highest weight set.

Contribution

It determines the exact weights and corresponding sequence sets for certain levels in the weight spectrum of binary Steinhaus triangles, advancing understanding of their structure.

Findings

Identified the set of weights for specific levels in the triangle spectrum.

Characterized the sequences generating triangles of these weights.

Provided partial results for the second-highest weight set.

Abstract

Let $\{0=w_0<w_1<w_2<\ldots<w_m\}$ be the set of weights of binary Steinhaus triangles of size $n$, and let $W_i$ be the set of sequences in $\mathbb{F}_2^n$ that generate triangles of weight $w_i$. We obtain the values of $w_i$ and the corresponding sets $W_i$ for $i\in\{1,2,3,m\}$, and partial results about $w_{m-1}$ and $W_{m-1}$.