### Syndrome Decoding Problem

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

Syndrome Decoding problem. Given integers $$n, k, w$$ such that $$k \leq n$$ and $$w \leq n$$, an instance of the problem $$\mathrm{SD}(n,k,w)$$ consists of a parity-check matrix $${\rm \bf H} \in \mathbb{F}_2^{(n-k) \times n}$$ and a vector $${\rm \bf s} \in \mathbb{F}_2^{n-k}$$ (called the syndrome). A solution to the problem is a vector $${\rm \bf e} \in \mathbb{F}_2^n$$ of Hamming weight $$\leq w$$ such that $${\rm \bf He^{\top}} = {\rm \bf s^{\top}}$$.

The challenge. Here, we focus on instances with code rate $$R = 0.5$$, that is $$n = 2k$$. We will choose a weight $$w$$ slightly higher than the Gilbert-Varshamov bound: $$w = \lceil 1.05 \, d_{\rm GV}\rceil$$. 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 $$\leq w$$ 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. The Low-weight Codeword challenge proposes another approach: instances of fixed cryptographic size but where the goal is to make $$w$$ as small as possible.

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 $$\leq w$$ 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
50059Greg Meyer
49059Greg Meyer
48059Greg Meyer
47058Greg Meyer
46055Greg Meyer

- lines 1, 3, 5, 7 are comments
- line 2 is the length $$n$$
- line 4 is the seed
- line 6 is the target weight $$w$$
- lines 8 to 7 + $$n/2$$ form a binary square matrix $${\rm \bf M}$$, such that $${\rm \bf H} = [ {\rm \bf I}_{n/2} | {\rm \bf M}^\top ]$$ is the parity-check matrix of the challenge
- line 8 + $$n/2$$ is a comment
- line 9 + $$n/2$$ is the target syndrome

Details here.

A list of instances with seed 0 (indexed by length)

Tooltips give an indication of complexity.