Generic erasure correcting sets: Bounds and constructions

A generic (r,m)-erasure correcting set generates for each binary linear code of codimension r a collection of parity check equations that enables iterative decoding of all potentially correctable erasure patterns of size at most m. As we have shown earlier, such a set essentially is just a parity check collection with this property for the Hamming code of codimension r . We prove non-constructively that for fixed m the minimum size F(r,m) of a generic (r,m)-erasure correcting set is linear in r . Moreover, we show constructively that F(r,3) 3(r − 1)log2 3 + 1, which is a major improvement on a previous construction showing that F(r,3) 1 + 2 r(r − 1). In the course of this work we encountered the following problem that may be of independent interest: what is the smallest size of a collection C ⊆ Fn2 such that, given any set of s independent vectors in Fn2, there is a vector c ∈ C that has inner product 1 with all of these vectors? We show non-constructively that, for fixed s, this number is linear in n. © 2006 Elsevier Inc. All rights reserved.