Low-density parity check codes (LDPC codes) were introduced by Gallager [2] and have received intensive study in the last few years, as advances in technology have made their implementation far more practical than when they were originally proposed. Basically, an LDPC code is a binary linear block code defined by a sparse m × n parity-check matrix H . Equivalently such a code can be thought of ...

A low-density parity-check code is a code specified by a parity-check matrix with the following properties : each column contains a small fixed numberj > 3 of I’s and each row contains a small fixed number k > j of 1’s. The typical minimum distance of these codes increases linearly with block length for a fixed rate and fixed j. When used with maximum likelihood decoding on a snfhciently quiet ...

The parity-check matrix of a linear code is used to define a bipartite code constraint (Tanner) graph in which bit nodes are connected to parity check nodes. The connectivity properties of this graph are analyzed using both local connectivity and the eigenvalues of the associated adjacency matrix. A simple lower bound on minimum distance of the code is expressed in terms of the two largest eige...

In this paper three outer bounds on the storage-repair bandwidth (S-RB) tradeoff of regenerating codes having parameter set {(n, k, d), (α, β)} under the exact-repair (ER) setting are presented. The tradeoff under the functionalrepair (FR) setting was settled in the seminal work of Dimakis et al. that introduced the framework of regenerating codes as well as a subsequent paper by Wu. While it i...

We construct various classes of low-density parity-check codes using point-line incidence structures in the classical projective plane PG(2, q). Each incidence structure is based on the various point classes (internal, external) and line classes (skew, tangent, secant) created by the geometry of a conic in the plane. For each class, we prove various properties about dimension and minimum distan...

In this paper, we discuss the reduction of errortrellises for tail-biting convolutional codes. In the case where some column of a parity-check matrix has a monomial factor D, we show that the associated tail-biting error-trellis can be reduced by cyclically shifting the corresponding error-subsequence by l (i.e., the power of D) time units. We see that the resulting reduced error-trellis is aga...

Using group theory, we analyze cycle GF(2) codes that use Cayley graphs as their associated graphs. First, we show that through row and column permutations the parity check matrix H can be put in a concatenation form of row-permuted block-diagonal matrices. Encoding utilizing this form can be performed in linear time and in parallel. Second, we derive a rule to determine the nonzero entries of ...

A Split decoding algorithm is proposed which divides each row of the parity check matrix into two or multiple nearly-independent simplified partitions. The proposed method significantly reduces the wire interconnect and decoder complexity and therefore results in fast, small, and high energy efficiency circuits. Three full-parallel decoder chips for a (2048, 1723) LDPC code compliant with the 1...

Moderate Density Parity Check (MDPC) codes are defined here as codes which have a parity-check matrix whose row weight is O( √ n) where n is the length n of the code. They can be decoded like LDPC codes but they decode much less errors than LDPC codes: the number of errors they can decode in this case is of order Ω( √ n). Despite this fact they have been proved very useful in cryptography for d...

Low Density Parity Check (LDPC) codes are one of the best error correcting codes that enable the future generations of wireless devices to achieve higher data rates with excellent quality of service. This paper presents two novel flexible decoder architectures. The first one supports (3, 6) regular codes of rate 1/2 that can be used for different block lengths. The second decoder is more genera...