Two-Dimensional Vector Linear Network Coding - Ma, Lan; Sun, Qifu Tyler; Li, Zongpeng (Proceedings / IEEE International Symposium on Information Theory, 2025)
In this work, we introduce a new framework of linear network coding (LNC) schemes called two-dimensional (2D) vector LNC, which subsumes conventional (1D) vector LNC as a special case. The new framework models every data unit as an L_{1} × L_{2} matrix, enabling a new encoding dimension so that data units can be coded along both row and column dimensions. Compared with (1D) vector LNC of block length L_{1} L_{2}, L_{1} × L_{2} 2D vector LNC not only reduces the size of local encoding kernels from L_{1} L_{2} × L_{1} L_{2} to L_{1} × L_{1} and L_{2} × L_{2} , but also exhibits greater scalability in the selection of local encoding kernels. In addition, we prove an equivalence between the linear solvability of L_{1} × L_{2} 2D vector LNC and (1D) vector LNC of block length L_{1} L_{2} . Under the framework of 2D vector LNC, we further formulate 2D circular-shift-based LNC. For distinct odd primes L_{1} and L_{2} , we prove that a network has a (1D) circular-shift-based linear solution of block length \left(L_{1}-1\right)\left(L_{2}-1\right) if and only if it has an \left(L_{1}-1\right) ×\left(L_{2}-1\right) 2D circular-shift-based linear solution. We also demonstrate through an explicit example that, compared with a (1D) circular-shift-based linear solution of block length 8, a 2 × 4 2D circular-shift-based linear solution requires 23 % fewer XORs in the coding process at intermediate nodes and 13 % fewer XORs in the decoding process.