Matrix structure, polynomial arithmetic, and erasure-resilient encoding/decoding

Authors:
Victor Y. Pan
Affiliations:
Mathematics and Computer Science Department, Lehman College, CUNY, Bronx, NY
Venue:
ISSAC '00 Proceedings of the 2000 international symposium on Symbolic and algebraic computation
Year:
2000

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Abstract

We exploit various matrix structures to decrease the running time and memory space of the known practical deterministic schemes for erasure-resilient encoding/decoding. Polynomial interpolation and multipoint evaluation enable both encoding and decoding in nearly linear time but the overhead constants are large (particularly, for interpolation), and more straightforward quadratic time algorithms prevail in practice. We propose faster algorithms. At the encoding stage, we decrease the running time per information packet from C log2 r, for a large constant C, or from r (for practical encoding) to log r. For decoding, our improvement is by the factors C and N/log N, respectively, for the input of size N. Our computations do not involve polynomial interpolation. Multipoint polynomial evaluation is either also avoided or is confined to decoding.