Superregular Lower Triangular Toeplitz Matrices for Low Delay Wireless Streaming
Jonas Hansen, Jan Stubbe Østergaard, Johnny Kudahl, John H. Madsen · IEEE Transactions on Communications · 2017
A matrix is termed superregular if all of its possible submatrices are non-singular. Superregular lower triangular Toeplitz matrices are useful for MDS convolutional codes and (sequential) network codes. In this paper, we present the explicit matrix constructions for superregular lower triangular Toeplitz matrices in GF(2p)k×k, k ≤ 5. For k > 5 we provide a greedy algorithm, which (over sufficiently large fields) is guaranteed to find a superregular lower triangular Toeplitz matrix. We introduce (product preserving) joint superregularity, and extend our explicit matrix constructions to these cases. We provide methods for deriving the exact symbol loss probability and delay for any deterministic block code. We derive the exact symbol loss probability and delay for codes using a superregular lower triangular matrix and for codes using two (product preserving) jointly superregular lower triangular matrices. We then compare these results with those obtained from both simulations and our practical implementation, and for each case we also compare with random-based codes. Furthermore, our experiments show a gain in coding throughput above 40% for superregular lower triangular Toeplitz matrices over random matrices.