Belief propagation decoding for short algebraic codes with permutations within the code space

A system for determining content from potentially corrupted data, the system comprising a controller configured to:

• receive data representing probabilities associated with a plurality of variables of a candidate codeword;
• receive a parity check matrix associated with a factor graph and corresponding to a codebook, the codebook comprising a plurality of valid codewords, the factor graph comprising variable nodes, check nodes and edges connecting variable and check nodes;
• calculate initial messages to be sent from variable nodes to associated check nodes based on the received data;
• and
• iteratively exchange messages between variable nodes and check nodes by iterating the following steps:
• sending messages from the variable nodes to associated check nodes;
• sending messages from the check nodes to associated variable nodes based on messages received at the check nodes;
• and
• after each exchange of messages between the variable nodes and check nodes:
• calculating messages based on messages received at the variable nodes;
• and
• calculating permuted messages to be sent from the variable nodes in the next iteration by applying a permutation to the calculated messages,
• wherein the permutation is of a set of permutations which map each codeword of the set of codewords onto a different codeword of the set of codewords;
• calculating an estimated codeword based on messages received by the variable nodes;
• applying to the estimated codeword the inverse of all permutations previously applied to determine content of the received data;
• and
• outputting the content.