Relaxation Bounds on the Minimum Pseudo-Weight of Linear Block Codes

Just as the Hamming weight spectrum of a linear block code sheds light on the performance of a maximum likelihood decoder, the pseudo-weight spectrum provides insight into the performance of a linear programming decoder. Using properties of polyhedral cones, we find the pseudo-weight spectrum of some short codes. We also present two general lower bounds on the minimum pseudo-weight. The first bound is based on the column weight of the parity-check matrix. The second bound is computed by solving an optimization problem. In some cases, this bound is more tractable to compute than previously known bounds and thus can be applied to longer codes.Comment: To appear in the proceedings of the 2005 IEEE International Symposium on Information Theory, Adelaide, Australia, September 4-9, 200

A tensor-product parity code for magnetic recording

In magnetic recording, a standard code architecture consists of an outer Reed–Solomon code in concatenation with an inner parity code. The inner parity code is used to detect and correct common error events. Generally, a parity code with short block length performs better, as multiple error events within one block and, consequently, miscorrection are less likely. In this paper, we study an inner code that offers the same system performance as a parity code with very short block length, even as short as the symbol length (in bits) of the outer Reed–Solomon code, but with higher code rate. This code is a tensor-product code, with a Bose–Chauduri–Hocquenghem (BCH) code and a short parity code as constituent codes. The decoder for this code is not much more complex than the optimal decoder of the baseline parity-coded channel; in fact, the only additional steps are Viterbi detection matched to the channel and decoding of the BCH code. Index Terms—BCH code, magnetic recording, parity code, Reed–Solomon code, tensor-product code. I

STABILIZATION OF BLOCK-TYPE-DECODABILITY PROPERTIES FOR CONSTRAINED SYSTEMS ∗

Abstract. We consider a class of encoders for constrained systems, which we call block-typedecodable encoders. For a constrained system presented by a deterministic graph, we design a blocktype-decodable encoder by selecting a subset of states of the graph to be used as encoder states. Such a subset is known as a set of principal states. Our goal is to find an optimal set of principal states, i.e., a set which yields the highest code rate. We study the relationship between optimal sets of principal states at finite block length and at asymptotically large block length. Specifically, we show that for a primitive constraint and a large enough block length, any optimal set of principal states is also asymptotically optimal. Moreover, we give bounds on the block length such that this relationship holds. We also characterize asymptotically optimal block-type-decodable encoders. Finally, we study the complexity of various problems related to block-type-decodable encoders

A Tensor-Product Parity Code for Magnetic Recording

In magnetic recording channels, a standard code architecture consists of an outer Reed-Solomon code in concatenation with an inner parity code. The inner parity code is used to detect and correct commonly occurring error events. Generally, a parity code with short block length performs better, as multiple error events within one block and, consequently, miscorrection are less likely

Design parameter optimization for perpendicular magnetic recording systems

In a perpendicular magnetic recording system, advanced read/write transducers, magnetic media, and signal processing techniques are combined to achieve the highest possible storage density, subject to severe constraints on reliability. This paper proposes a quasi-analytic methodology for exploring the complex design tradeoffs among these system components. We use a simple channel model, characterized by three parameters: isolated voltage pulse width, transition jitter noise variance, and additive electronic/replay head noise power. The system incorporates generalized partial-response equalization and maximum-likelihood detection, along with a Reed–Solomon error-correcting code characterized by its code rate. We calculate a family of “design curves ” from which we can determine, for a given set of channel parameters, the maximum user density that can be achieved with a specified codeword error rate, along with the corresponding code rate. The design curves can also be used to determine the acceptable range of channel parameters consistent with a target user density and codeword error rate. Index Terms—Additive noise, jitter noise, parameter optimization, perpendicular recording, Reed–Solomon code, Viterbi detector. I