On the Exact Lower Bounds of Encoding Circuit Sizes of Hamming Codes and Hadamard Codes

TL;DR

This paper establishes the exact minimum circuit sizes needed for encoding Hamming and Hadamard codes, providing optimal encoding algorithms that meet these bounds.

Contribution

It proves the precise lower bounds for encoding circuit sizes of Hamming and Hadamard codes and presents algorithms that achieve these bounds.

Findings

Exact lower bounds for encoding circuit sizes are established.

Encoding algorithms are developed to meet these bounds.

Optimal encoding schemes for Hamming and Hadamard codes are demonstrated.

Abstract

In this paper, we investigate the encoding circuit size of Hamming codes and Hadamard codes. To begin with, we prove the exact lower bound of circuit size required in the encoding of (punctured)~Hadamard codes and (extended)~Hamming codes. Then the encoding algorithms for (punctured)~Hadamard codes are presented to achieve the derived lower bounds. For (extended)~Hamming codes, we also propose encoding algorithms that achieve the lower bounds.