Efficient encoding of cycle codes: a graphical approach

The paper presents a low-complexity encoding algorithm for cycle codes-low-density parity-check (LDPC) codes with column weight j = 2. For a cycle code of block length n, the encoding complexity of the algorithm is O(n) whereas the conventional encoding method has a complexity of O(n²). We achieve the linear-complexity encoder through graph analysis. First, we introduce particular Tanner graphs named pseudo-trees and prove that any pseudo-tree can be encoded in linear time. Second, we show that any cycle code can be represented by the union of pseudo-trees. We show that by combining these two ideas, cycle codes are linear-time encodable.