The NASA proposal for the Consultative Committee for Space Data Systems (CCSDS) experimental specification CCSDS 131.1-O-2 [1] describes a set of low-density parity-check (LDPC) codes for nearEarth and deep-space applications. That experimental specification describes two kinds of codes. The second one is a set of accumulate–repeat–4–jagged–accumulate (AR4JA) codes, which have characteristics p...

We prove that for every prime p ≤ poly(n) there exists a (0, 1)-matrix M of size tp(n,m) × n where tp(n,m) = O ( m+ m log n m log min(m, p) ) such that every m columns of M are linearly independent over Zp, the field of integers modulo p (and therefore over any field of characteristic p and over the real numbers field R). In coding theory this matrix is a parity-check (0, 1)-matrix over Zp of a...

We consider the possibility of extending to a family of sets a binary set function defined on a subfamily so that the extension is, in fact, uniquely determined. We place in this context the problem of finding the least integer n(r) such that every linear code of length n with n B n(r), dimension n r and minimum Hamming distance at least 4 has a parity check matrix composed entirely of odd weig...

In this paper we give a comparative analysis of decoding algorithms of Low Density Parity Check (LDPC) codes in a channel with the Nakagami distribution of the fading envelope. We consider the Progressive Edge-Growth (PEG) method and Improved PEG method for the parity check matrix construction, which can be used to avoid short girths, small trapping sets and a high level of error floor. A compa...

An important property of high-performance, low complexity codes is the existence of highly efficient algorithms for their decoding. Many of the most efficient, recent graph-based algorithms, e.g. message passing algorithms and decoding based on linear programming, crucially depend on the efficient representation of a code in a graphical model. In order to understand the performance of these alg...

In this paper, a method to design regular (2, dc)-LDPC codes over GF(q) with both good waterfall and error floor properties is presented, based on the algebraic properties of their binary image. First, the algebraic properties of rows of the parity check matrix H associated with a code are characterized and optimized to improve the waterfall. Then the algebraic properties of cycles and stopping...

This paper presents a construction of low-density parity-check (LDPC) codes based on the incidence matrices of oval designs. The new LDPC codes have regular parity-check matrices and Tanner graphs free of 4-cycles. Like the finite geometry codes, the codes from oval designs have parity-check matrices with a large proportion of linearly dependent rows and can achieve significantly better minimum...

This paper presents a construction of very high-rate low-density parity-check (LDPC) codes based on the incidence matrices of unital designs. Like the projective geometry and oval designs, unital designs exist with incidence matrices which are significantly rank deficient. Thus high-rate LDPC codes with a large number of linearly dependent parity-check equations can be constructed. The LDPC cod...

This paper introduces a class of linear codes which are non-uniform error correcting, i.e. they have the capability of correcting different errors in different codewords. A technique for specifying error characteristics in terms of algebraic inequalities, rather than the traditional spheres of radius e, is used. A construction is given for deriving these codes from known linear block codes. Thi...

In this paper, we present a lower bound for low density closed loop burst error correcting code in two sub-blocks of length 1 n and 2 n of the total code length 1 2 ( ) n n n n   . Keywords-Open loop burst, Closed loop burst, Parity check matrix,Syndromes, Error correction.