The Biased Long Code and Hardness of Vertex Cover

Abstract

5.1 Biased Long Code While the biased long code can be viewed as an encoding scheme, it is more con­ venient to take a combinatorial view and treat it as a weighted Kneser graph. Valid codewords correspond to certain large (in fact the largest) independent sets in this graph. Definition 5.1 For a bias parameter p ∈ (0, 1) and alphabet Σ, the vertex set of weighted Kneser graph Gp[Σ] is P(Σ), the family of all subsets of Σ. The weight of a vertex A ⊆ Σ is μp(A) = p|A|(1 − p)|Σ|−|A|. The edge set is { (A, B) | A, B ⊆ Σ, A ∩ B = φ}. For a family F ⊆ P(Σ), let μp(F ) denote its weight under μp.