# Maximum sum rank distance codes

This page lists some maximal sum rank distance codes with various lengths and the matrix size 2\*2. For the constructions of MSRD codes, we refer to the following references.

1. U. MartÍnez-Peňas and S. Puchinger, Maximum sum-rank distance codes over finite chain rings, IEEE Transactions on Information Theory, early access, 2024.  

2. U. MartÍnez-Peňas, New constructions of MSRD codes, arXiv:2402.03084, 2024.

3. H. Chen, New explicit linear sum-rank-metric codes, IEEE Transactions on Information Theory, vol. 69, no. 10, pp. 6303-6313, 2023.

4. Martínez-Peñas U. Skew and linearized Reed–Solomon codes and maximum sum rank distance codes over any division ring, Journal of Algebra, vol. 504, pp. 587-612, 2018.

5. Neri A. Twisted linearized Reed-Solomon codes: A skew polynomial framework. Journal of Algebra, vol. 609, pp. 792-839, 2022. 

6. H. Lao, Y.M. Chee, H. Chen and V. K. Vu, New constructions for linear maximum sum-rank Distance codes, submitted, 2023.

7. E. Byrne, H. Gluesing-Luerssen and A. Ravagnani, Fundamental properties of sum-rank-metric codes, IEEE Transactions on Information Theory, vol. 67, no. 10, pp. 6456-6475, 2021.

8. **Comparison of the parameters of some F\_q-linear MSRD codes**

<table><thead><tr><th width="133">Constructions</th><th>Block length t</th><th>n_1,...,n_t</th><th width="175">m_1,...,m_t</th><th>d_sr</th><th>k</th></tr></thead><tbody><tr><td>VII.1,[7]</td><td>&#x3C;=q^m+1</td><td>1</td><td>m</td><td>&#x3C;=t</td><td>m(t-d_sr+1)</td></tr><tr><td>VII.2,[7]</td><td>-</td><td>-</td><td>-</td><td>2</td><td>∑_{i=2}^{t}m_in_i+m_1(n_1-1)</td></tr><tr><td>VII.3a,[7]</td><td>-</td><td>-</td><td>-</td><td>∑_{i=1}^{t}n_i</td><td>m_t</td></tr><tr><td>VII.4,[7]</td><td>t=t_1+t_2,t_2≤q+1</td><td>n_{t_1+1},...,n_t=1</td><td>m_{t_1+1},...,m_t=1</td><td>∑_{i=1}^{t_1}n_i+t_2-m_{t_1}+1</td><td>m_{t_1}</td></tr><tr><td>VII.5,[7]</td><td>t=t_1+t_2,t_2≤q^{\hat{m}}+1</td><td>n_{t_1+1},...,n_t=1</td><td>m_{t_1}=\hat{m}a,m_{t_1+1},...,m_t=\hat{m}</td><td>∑_{i=1}^{t_1}n_i+t_2-a+1</td><td>m_{t_1}</td></tr><tr><td>VII.6,[7]</td><td>t=s+m+2,s≤m+m(m-1)/2+1</td><td>1</td><td>m_i=m for i∈[s+1],m_i=1 for i∈[s+2,s+m+2]</td><td>s+2</td><td>m+1</td></tr><tr><td>Theorem 5.2 [3]  </td><td>-</td><td>m_i=n_i for i∈[1,t]</td><td>m_{J-1}≥k,m_J≥∑<em>{J+1}^t m_i^2</em></td><td>∑_<em>{i=1}^{J-1}n_i+</em>β,J∈[1,t],β∈[1,n_J]</td><td>∑_<em>{i=J}^{t}m_i^2-m_J(</em>β-1)</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Theorem 4,[4]</td><td>t≤q-1</td><td>-</td><td>m</td><td>d_{sr}</td><td>m(N-d_{sr}+1)</td></tr><tr><td>Theorem 6.3, [5]</td><td>t≤q-1,(-1)^{kN}N_{ F_<em>{q^m}/ F_</em>{q}}(η)</td><td>m</td><td>m</td><td>d_{sr}</td><td>m(N-d_{sr}+1)</td></tr></tbody></table>