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?**

- 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.
- 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 \( {\rm \bf He^{\top}} = {\rm \bf s^{\top}} \).
- 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.

Length | Weight | Authors | Details |
---|---|---|---|

80 | 79 | Matthieu Lequesne | See details |

70 | 69 | Matthieu Lequesne | See details |

60 | 59 | Matthieu Lequesne | See details |

50 | 49 | Matthieu Lequesne | See details |

40 | 39 | Matthieu Lequesne | See details |

Submit your solution

Hall of fame

##### Download instances

Instance generator

Hall of fame

- lines 1, 3, 5, 7 are comments

- line 2 is the length \( n \)

- line 4 is the seed

- line 6 is the dimension \( k \)

- line 8 is the target weight \( w \)

- lines 10 to 10 + \( k \) form a ternary
matrix \( {\rm \bf M} \), such that
\( {\rm \bf H} = [ {\rm \bf I}_{n-k} | {\rm \bf M}^\top ] \)
is the parity-check matrix of the challenge

- the last line is the target syndrome

Details here.

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

Tooltips give an indication of complexity.