Plotkin construction: rank and kernel

TL;DR

This paper investigates the Plotkin construction for binary codes, demonstrating that the rank and kernel dimension of the resulting code are additive properties of the initial codes, applicable to both linear and nonlinear cases.

Contribution

It proves that the rank and kernel dimension of codes obtained via Plotkin construction are sums of the original codes' parameters, extending understanding to nonlinear codes.

Findings

Rank of the constructed code equals the sum of the ranks of the original codes.

Kernel dimension of the constructed code equals the sum of the kernels of the original codes.

Results apply to both linear and nonlinear binary codes.

Abstract

Given two binary codes of length n, using Plotkin construction we obtain a code of length 2n. The construction works for linear and nonlinear codes. For the linear case, it is straightforward to see that the dimension of the final code is the sum of the dimensions of the starting codes. For nonlinear codes, the rank and the dimension of the kernel are standard mesures of linearity. In this report, we prove that both parameters are also the sum of the corresponding ones of the starting codes.