New bounds and constructions for multiply constant-weight codes

TL;DR

This paper investigates bounds and constructions for multiply constant-weight codes (MCWCs), providing improved upper bounds, asymptotic existence results, and explicit constructions for certain parameters to enhance their reliability in physical unclonable functions.

Contribution

It introduces new upper bounds, proves the asymptotic optimality of certain MCWCs, and constructs specific optimal codes with total weight four and distance six.

Findings

New upper bounds improve previous Johnson-type bounds.

Asymptotic existence of certain optimal MCWCs established.

Explicit constructions for MCWCs with total weight four and distance six.

Abstract

Multiply constant-weight codes (MCWCs) were introduced recently to improve the reliability of certain physically unclonable function response. In this paper, the bounds of MCWCs and the constructions of optimal MCWCs are studied. Firstly, we derive three different types of upper bounds which improve the Johnson-type bounds given by Chee {\sl et al.} in some parameters. The asymptotic lower bound of MCWCs is also examined. Then we obtain the asymptotic existence of two classes of optimal MCWCs, which shows that the Johnson-type bounds for MCWCs with distances $2\sum_{i=1}^mw_i-2$ or $2mw-w$ are asymptotically exact. Finally, we construct a class of optimal MCWCs with total weight four and distance six by establishing the connection between such MCWCs and a new kind of combinatorial structures. As a consequence, the maximum sizes of MCWCs with total weight less than or equal to four are determined almost completely.