Large Weight Syndrome Decoding Problem

This page is dedicated to the Large Weight Ternary Syndrome Decoding problem for random binary linear codes.

Large Weight Ternary Syndrome Decoding problem. Given integers \(n, k, w\) such that \(k \leq n\) and \( w \leq n\), an instance of the problem \(\mathrm{LW3SD}(n,k,w)\) consists of a parity-check matrix \({\rm \bf H} \in \mathbb{F}_3^{(n-k) \times n}\) and a vector \({\rm \bf s} \in \mathbb{F}_3^{n-k}\) (called the syndrome). A solution to the problem is a vector \( {\rm \bf e} \in \mathbb{F}_3^n \) of Hamming weight \(\geq w\) such that \( {\rm \bf He^{\top}} = {\rm \bf s^{\top}} \).

The challenge. Here, we focus on instances with code rate \( R = \log_3(2) \simeq 0.36907 \), that is \(k = \lfloor R n \rfloor\). At this rate, the equivalent of the Gilbert-Varshamov bound for heigh weight words is exactly equal to \(n\). Hence we will take : \(w = \lfloor 0.99 \, d_{\rm GV}^{+}\rfloor = \lfloor 0.99 \, n \rfloor\). The matrix \( {\rm \bf H} \) and the syndrome \( {\rm \bf s} \) are picked uniformly at random. In this context, with very high probability there exists a vector \( {\rm \bf e} \) of weight \(\geq w \) such that \( {\rm \bf He^{\top}} = {\rm \bf s^{\top}} \).

Under these conditions, instances with cryptographic size are assumed to be out of reach, so we propose instances with increasing size to see how hard this problem is in practice.

Instance generation. The instances are generated using a Python script. This script takes as input the length of the code and a seed.

How to participate?

1. Choose the value of \(n\) for which you want to solve the problem and download the corresponding instance (on the right). You can also use the generator to generate an instance with another random seed.
2. Parse the instance to get the values of \(w, \rm \bf H\) and \(\rm \bf s\). Find a vector \(\rm \bf e\) of weight \(\geq w\) such that \( {\rm \bf He^{\top}} = {\rm \bf s^{\top}} \).
3. Submit your solution using the submission form. If your solution is correct and you solved a new challenge, your name will appear in the hall of fame.

Best solutions

Length	Weight	Authors	Details
200	198	Andre Esser, Alexander May and Floyd Zweydinger	See details
190	189	Andre Esser, Alexander May and Floyd Zweydinger	See details
180	178	Andre Esser, Alexander May and Floyd Zweydinger	See details
170	168	Andre Esser, Alexander May and Floyd Zweydinger	See details
160	158	Andre Esser, Alexander May and Floyd Zweydinger	See details

Submit your solution
Hall of fame

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Download instances

Instance generator Format of instances