[Paper Review] Constructions of Optimal and Near-Optimal Multiply Constant-Weight Codes

Yeow Meng Chee, Han Mao Kiah+2 more|arXiv (Cornell University)|Nov 10, 2014

Physical Unclonable Functions (PUFs) and Hardware Security20 references3 citations

TL;DR

This paper presents combinatorial constructions for optimal and near-optimal multiply constant-weight codes (MCWCs), establishing that Johnson-type upper bounds are asymptotically tight for fixed weights and distances. It introduces generalized packing designs to fully determine optimal MCWC sizes for total weight four and distance four, and constructs infinite families of optimal MCWCs with weight two or three per part using resolvable BIBDs and product constructions.

ABSTRACT

Multiply constant-weight codes (MCWCs) have been recently studied to improve the reliability of certain physically unclonable function response. In this paper, we give combinatorial constructions for MCWCs which yield several new infinite families of optimal MCWCs. Furthermore, we demonstrate that the Johnson type upper bounds of MCWCs are asymptotically tight for fixed weights and distances. Finally, we provide bounds and constructions of two dimensional MCWCs.

Motivation & Objective

• To construct infinite families of optimal and near-optimal multiply constant-weight codes (MCWCs) with small weights and distances.
• To determine the maximum size of MCWCs with total weight four and minimum distance four using generalized packing designs.
• To establish the asymptotic tightness of Johnson-type upper bounds for fixed weights and distances in MCWCs.
• To introduce and analyze two-dimensional MCWCs with row and column weight constraints, providing bounds and constructions.
• To extend existing results on MCWCs by improving the asymptotic tightness of upper bounds from a constant factor to exact asymptotic equivalence.

Proposed method

• Leverages generalized packing designs—recently studied by Bailey and Burgess—to model and analyze MCWCs with total weight four.
• Applies product constructions to scale small optimal 2D MCWCs into larger optimal codes by concatenating codewords vertically or horizontally.
• Uses α-resolvable balanced incomplete block designs (BIBDs) to construct optimal 2D MCWCs, where parameters are derived from block design existence conditions.
• Employs fractional matching theorems (Kahn, 2001) to prove that Johnson-type upper bounds for MCWCs are asymptotically tight for fixed weights and distances.
• Introduces the concept of two-dimensional MCWCs with row and column weight constraints, modeling them as derived regular packings (DRPs).
• Derives bounds on the maximum size of 2D MCWCs using combinatorial inequalities based on DRP parameters and applies these to validate optimality.

Experimental results

Research questions

• RQ1 What is the maximum size of an MCWC with total weight four and minimum distance four?
• RQ2 Can infinite families of optimal MCWCs be constructed for weight two or three per part and minimum distance four?
• RQ3 Are Johnson-type upper bounds for MCWCs asymptotically tight for fixed weights and distances?
• RQ4 What are the necessary and sufficient conditions for the existence of optimal two-dimensional MCWCs with given row and column weights?
• RQ5 How can product constructions and resolvable BIBDs be used to generate optimal 2D MCWCs from smaller optimal codes?

Key findings

• The maximum size of MCWCs with total weight four and minimum distance four is completely determined using generalized packing designs, except for one small unresolved case.
• Infinite families of optimal MCWCs with weight two or three per part and minimum distance four are constructed using large sets of optimal packings and product techniques.
• Johnson-type upper bounds for MCWCs are asymptotically tight for fixed weights and distances, improving upon prior results that only established tightness to within a constant factor.
• The existence of α-resolvable BIBDs with specific parameters implies the existence of optimal two-dimensional MCWCs with corresponding row and column weights.
• A Latin square of order $ n $ achieves the upper bound for $ M(n,n,2n,1,1) = n $, confirming optimality for this class of 2D MCWCs.
• Product constructions preserve optimality when concatenating codewords in vertical or horizontal directions, enabling scaling of optimal codes to larger parameters.