Systems and methods for advanced iterative decoding and channel estimation of concatenated coding systems

A method for modifying variable-to-check messages during an iteration in decoding of codes represented by a parity check matrix, the method comprising the following steps for at least one variable node:

• a. calculating variable-to-check messages Mvc(i,j)(n-1) and Mvc(i,j)(n) from variable node j to check node i, respectively, in iterations (n−1) and (n), where n>2;
• b. comparing the positive or negative signs of variable-to-check messages Mvc(i,j)(n-1) and Mvc(i,j)(n) calculated in step a;
• and
• c. if said signs in step b) are different, generate modified variable-to-check message {acute over (M)}vc(i,j)(n) according to {acute over (M)}vc(i,j)(n)=g·Mvc(i, j)(n)+(1−g)·Mvc(i,j)(n-1), where 0.5<g≤1.

A method for simple greedy scheduling of check equation updates, for at least M equations, 1<M≤N−K, during an iteration in decoding of codes represented by a parity check matrix with N−K parity check rows and N columns, comprising the following steps:

• a. for the set of M′≤M of non-updated check nodes out of M check nodes of the parity check matrix, calculating Vali=Min1+Min2, i=1, 2, . . . , L where L≥1 and where Min1 and Min2 are the two smallest values in the set of absolute values of variable-to-check messages {|Mvc(i, :)|} where index i corresponds to the set of non-updated check nodes;
• b. sorting the set {Vali} calculated in step a in decreasing order to obtain an ordering vector I={I1, I2, . . . , IL}, such that I1 is the index of a check node with the largest value Val, I2 is the index of a check node with the next largest value Val and IL, is the index of a check node with the smallest value Val;
• c. updating L check node equations selected in step b according to the ordering vector, I={I1, I2, . . . , IL} calculated in step b, by calculating and propagating corresponding check-to-variable messages and updating variable-to-check messages for all variables that receive check-to-variable messages in this step;
• and
• d. repeating steps a, b and c until all check nodes are updated by calculating and propagating corresponding check-to-variable messages.

A method for decoding codes represented by a parity check matrix of dimension (N−K)×N, comprising:

• a. generating P>1 parity check matrices with N−K sparse columns wherein up to N−K sparse columns contain only a single entry equal to 1 per column, wherein sparse columns of each of P parity check matrices correspond to different subsets of N−K bit log-likelihood ratios of N−K+R least reliable bit log-likelihood ratios, wherein R≥P is a configurable integer;
• b. decoding of channel log-likelihood ratios using said P parity check matrices with sparse columns to produce updated log-likelihood ratios by employing soft-input soft-output message passing decoding until a desired number of iterations is reached or until said decoding using at least one of said P matrices produces a valid codeword;
• and
• c. on the condition that decoding using said P parity check matrices produces no valid codeword, performing additional decoding that is based at least in part on algebraic decoding of the sequences of said updated log-likelihood ratios.

A method for simple greedy scheduling of check equation updates, for at least M equations, where 1<M≤N−K, during an iteration in decoding of codes represented by a parity check matrix with N−K parity check rows and N columns, the method comprising the following steps:

• a. for each check node i, out of M check nodes of the parity check matrix, calculating Vali=Min1+Min2, i=1, 2, . . . , L where L≥1 and where Min1 and Min2 are the two smallest values in the set of absolute values of variable-to-check messages {|Mvc(i,:)|} where index i corresponds to the set of check nodes;
• b. sorting the set {Vali} calculated in step a in decreasing order to obtain an ordering vector I={I1, I2, . . . , IM}, such that I1 is the index of a check node with the largest value Val, I2 is the index of a check node with the next largest value Val and IM is the index of a check node with the smallest value Val;
• and
• c. updating M check node equations, according to the ordering vector, I={I1, I2, . . . IM} calculated in step b, by calculating and propagating corresponding check-to-variable messages.