Constant Weight Codes: A Geometric Approach Based on Dissections

TL;DR

This paper introduces a geometric method for encoding and decoding constant weight binary codes by embedding the codebook in Euclidean space and using dissections to establish bijections, improving efficiency for small weights.

Contribution

It presents a novel geometric dissection algorithm for constant weight codes, with complexity depending on weight rather than block length, offering efficiency advantages.

Findings

Algorithm is correct and efficient for small weights

Complexity depends on weight, not block length

Advantageous when weight is less than square root of block length

Abstract

We present a novel technique for encoding and decoding constant weight binary codes that uses a geometric interpretation of the codebook. Our technique is based on embedding the codebook in a Euclidean space of dimension equal to the weight of the code. The encoder and decoder mappings are then interpreted as a bijection between a certain hyper-rectangle and a polytope in this Euclidean space. An inductive dissection algorithm is developed for constructing such a bijection. We prove that the algorithm is correct and then analyze its complexity. The complexity depends on the weight of the code, rather than on the block length as in other algorithms. This approach is advantageous when the weight is smaller than the square root of the block length.