Structured iteratively decodable codes based on Steiner systems and their application in magnetic recording

Abstract

This paper introduces a combinatorial construction of a class of iteratively decodable codes, an approach diametrically opposed to the prevalent practice of using large, random-like codes. Our codes are wellstructured and, unlike random codes, can lend themselves to a very low complexity implementation. A systematic way of constructing codes based on Steiner systems and the Z Z v , group is presented, and a hardware efficient encoding algorithm is proposed. A substantial performance improvement of high-rate Steiner codes over the existing schemes used in magnetic recording systems is demonstrated. .

TS-LDPC Codes: Turbo-Structured Codes With Large Girth

“…9, the rightmost digits of , , , are also the same. Hence, by the above reasoning (11) By (11), we derive that (12) Combining (10) and (12), we derive that (13) Since and are connected to the same check node , and are different only in the rightmost digit by definition. Therefore, all the digits of are zeros except for the rightmost one.…”

“…Therefore, all the digits of are zeros except for the rightmost one. Since by (13), should also have all zero digits except for the rightmost digit. Hence, and are different only in the rightmost digit, which implies that the two auxiliary nodes and are children of the same check node.…”

“…Finite-geometry LDPC codes have girth . Another method to construct structured -cycle-free regular LDPC codes is based on balanced incomplete block designs (BIBD) [9]- [13]. A BIBD is defined as a collection of equal size blocks, comprising elements drawn from a set , such that each pair of distinct elements of occurs in exactly blocks of .…”

Generalization of Tanner's minimum distance bounds for LDPC codes

“…described by the associated polynomial h ij (x) corresponding to the top row of H ij [8], [9], [10]. More precisely, we have…”

The design of structured regular LDPC codes with large girth

“…Kou, Lin, and Fossorier [5] construct four classes of (quasi-)cyclic LDPC codes with girth 6 based on finite geometries of lines and points of Euclidean and projective geometries over finite fields. References [6,7] present LDPC codes with column weight 3 based on balanced incomplete block designs (BIBD) with girth 6, i.e, with no cylces of lenght 4. Here we construct codes with girth up to 20, much larger than all these codes described in the literature.…”