Universal error control coding scheme for digital communication and data storage systems

A method for providing forward error-correction coding for data that is transmitted from a source to a destination over a communication medium, comprising:

• encoding message blocks, u, of said data to codewords, v, as follows:
• v=uG, where G is a generator matrix;
• and
• using a Quasi-Cyclic Low Density Parity Check Code to implement a parity check matrix H, such that every vector in the row space of the generator matrix G is orthogonal to the rows of parity check matrix H and parity check matrix H is given as an array of sparse circulant matrices of the same size comprising:
• defining said selected Quasi-Cyclic Low Density Parity Check Code Cqc as the null space of a parity check matrix Hqc, which is a c×t array of b×b circulants over the binary Galois field GF(2) of the form given by:
• H
• qc
• =
• [
• A
• 1
• ,
• 1
• A
• 1
• ,
• 2
• …
• A
• 1
• ,
• t
• A
• 2
• ,
• 1
• A
• 2
• ,
• 2
• …
• A
• 2
• ,
• t
• ⋮
• ⋮
• ⋱
• ⋮
• A
• c
• ,
• 1
• A
• c
• ,
• 2
• …
• A
• c
• ,
• t
• ]
• ;
• choosing any sequence of m distinct pairs of positive integers, (ci, ti), with 1≦i≦m, such that 1≦c1≦c2≦ . . . ≦cm≦c, and 1≦t1≦t2≦ . . . ≦tm≦t;
• let Hqc(ci, ti) be the ci×ti subarray of the array Hqc of circulants of said selected Quasi-Cyclic Low Density Parity Check Code obtained by deleting the last t−ti columns and the last c−ci rows of b×b circulant matrices;
• and
• let C(ci, ti) be the null space of the matrix Hqc(ci, ti) for each of said sequence of m distinct pairs of positive integers, forming a member of said structurally compatible family Cqc, C(ci, ti), . . . , C(cm, tm) of codes for each of said sequence of m distinct pairs of positive integers.