An Explicit Construction of Universally Decodable Matrices
<|reference_start|>An Explicit Construction of Universally Decodable Matrices: Universally decodable matrices can be used for coding purposes when transmitting over slow fading channels. These matrices are parameterized by positive integers $L$ and $n$ and a prime power $q$. Based on Pascal's triangle we give an explicit construction of universally decodable matrices for any non-zero integers $L$ and $n$ and any prime power $q$ where $L \leq q+1$. This is the largest set of possible parameter values since for any list of universally decodable matrices the value $L$ is upper bounded by $q+1$, except for the trivial case $n = 1$. For the proof of our construction we use properties of Hasse derivatives, and it turns out that our construction has connections to Reed-Solomon codes, Reed-Muller codes, and so-called repeated-root cyclic codes. Additionally, we show how universally decodable matrices can be modified so that they remain universally decodable matrices.<|reference_end|>
@article{vontobel2005an,
    title={An Explicit Construction of Universally Decodable Matrices},
    author={Pascal O. Vontobel, Ashwin Ganesan},
    journal={arXiv preprint arXiv:cs/0508098},
    year={2005},
    archivePrefix={arXiv},
    eprint={cs/0508098},
    primaryClass={cs.IT cs.DM math.IT}
}