Equivalent mean breakdown points for linear codes and compressed sensing by ℓ1 optimization

Decoding a linear code consists in recovering an input vector x ∈ ℝⁿ from corrupted oversampled measurements y = Bx + w where B ∈ ℝ^m×n is a full rank matrix and the noise w ∈ ℝ^m is a sparse random vector. If n ≤ m/2, decoding can be done directly. However, if n > m/2, to save computing time, this problem can be transformed into an underdetermined compressed sensing problem, Aw = :z = Ay, for the syndrome z by means of a full rank matrix A ∈ ℝ^d×m, d = m - n, such that AB = 0. For this purpose, to have equivalently high mean breakdown points by ℓ₁ linear programming, we use uniformly distributed random matrices A ∈ ℝ^(m-n)×m and matrices B ∈ ℝ^m×n with orthonormal columns spanning the null space of A. The numerical results are collected in figures and tables.