Finite-length analysis of low-density parity-check codes on the binary erasure channel
IEEE Transactions on Information Theory
On the stopping distance and the stopping redundancy of codes
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Single-exclusion number and the stopping redundancy of MDS codes
IEEE Transactions on Information Theory
Stopping set distributions of algebraic geometry codes from elliptic curves
TAMC'12 Proceedings of the 9th Annual international conference on Theory and Applications of Models of Computation
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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)log23 + 1, which is a major improvement on a previous construction showing that F(r, 3) ≤ 1 + 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 ⊆ F2n such that, given any set of s independent vectors in F2n, 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.