System and method for error correction of digitized phase signals from MR/GMR head readback waveforms

A system and method for algebraically correcting errors in complex digitized phase signals from a magneto-resistive or giant magneto-resistive (MR/GMR) head readback waveform includes a data state machine that encodes phase symbols into data bits in accordance with, e.g., the (1, 10) constraint and a parity state machine that generates parity symbols such that a single inserted parity symbol does not violate the (1, 7) constraint in a run length limited code and furthermore the data following the insertion will not violate the (1, 10) constraint in a run length limited code. The state machines can be used as a trellis to perform maximum likelihood decoding on received coded data, thus performing soft algebraic error detection on received data. The invention thus guarantees better overall error rate performance than hard decision post processing of blocks of detected bits by a parity check matrix which is otherwise vulnerable to loss of bit synchronization at high linear density recording.