Precoding Mapping Optimization for Magnetic Recording Channels

We study the problem of designing a rate-1 block-precoder to minimize bit/symbol error rate when storing a given source on a magnetic recording channel. A block-precoder of length b-bits is defined by a permutation π on 2b blocks. We show that the problem of finding a permutation for the block-precoder that minimizes bit/symbol error rate is equivalent to solving the quadratic assignment problem, a known combinatorial optimization problem that is NP-complete. We exploit the symmetry group of the b-dimensional hypercube to reduce the search space, allowing a branch-and-bound technique to find the optimal 5-bit precoders. We also implement a local search algorithm that can find good precoders for larger blocklengths. We design precoders for MTR-constrained user bits and unconstrained parity bits with a reverse-concatenation architecture, and we evaluate the resulting SNR gains in a turbo equalization scheme.