On the Complexity of finding stopping set size in Tanner Graphs

Abstract

Two decision problems related to the computation of stopping sets in Tanner graphs are shown to be NP-complete. NP-hardness of the problem of computing the stopping distance of a Tanner graph follows as a consequence.

Permutation Decoding and the Stopping Redundancy Hierarchy of Linear Block Codes

“…Example 3.3: Let G be the [24,12,8] extended Golay code. The code G is self-dual and contains a [24,10,8] subcode.…”

“…The cog indexed by A has the following octal representation cog [23,12] Upper bounds U (ρ j (C)) on the stopping redundancy hierarchy obtained from cogs cog [23,12],A , cog [23,12],B , and cog [23,12],D of the parity-check matrix of the [23,12,7] Golay code and the general upper bounds obtained from Theorem 3.9 with ǫ = 10…”

“…The matrix was constructed by identifying the cog of the [23,12,7] code that leads to optimal stopping distance properties, generating a m = 21 cyclic parity-check matrix, and then adding a single (fixed) parity-check position to each of the rows.…”

“…H [24,12] 1 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 0 1 1 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 1 1 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 1 1 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 1 0 1 1 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 0 1 1 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 1 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 …”

“…In Table X we list the number of uncorrectable erasure patterns of size up to σ = 12 in several parity-check matrices of the [24,12,8] extended Golay code; H [24,12],⋆ and H W were defined in Section V, while the matrix H HS corresponds to the matrix of dimension 34 × 24 described in [4]. The index ML refers to the erasure patterns that cannot be decoded by an ML …”