# Signature codes for weighted binary adder channel and multimedia fingerprinting

## TL;DR

This paper investigates the theoretical limits and constructions of signature codes for weighted binary adder channels and multimedia fingerprinting, providing bounds, exact values, and new code designs with applications to noisy environments.

## Contribution

It derives tight bounds, determines exact code sizes for specific cases, and introduces new signature code constructions with efficient decoding for multimedia fingerprinting.

## Key findings

- Upper bounds relate signature codes to bipartite graphs and $B_t$ codes.
- Exact values of $A(n,2,2)$ and $A(n,3,2)$ are established for many $n$.
- New code constructions enable efficient decoding and are applicable to noisy scenarios.

## Abstract

In this paper, we study the signature codes for weighted binary adder channel (WbAC) and collusion-resistant multimedia fingerprinting. Let $A(n,t)$ denote the maximum cardinality of a $t$-signature code of length $n$, and $A(n,w,t)$ denote the maximum cardinality of a $t$-signature code of length $n$ and constant weight $w$. First, we derive asymptotic and general upper bounds of $A(n,t)$ by relating signature codes to $B_t$ codes and bipartite graphs with large girth respectively, and also show the upper bounds are tight for certain cases. Second, we determine the exact values of $A(n,2,2)$ and $A(n,3,2)$ for infinitely many $n$ by connecting signature codes with $C_4$-free graphs and union-free families, respectively. Third, we provide two explicit constructions for $t$-signature codes which have efficient decoding algorithms and applications to two-level signature codes. Furthermore, we show from the geometric viewpoint that there does not exist any binary code with complete traceability for noisy WbAC and multimedia fingerprinting. A new type of signature codes with a weaker requirement than complete traceability is introduced for the noisy scenario.