Product codes are widely used in data-storage, optical and wireless applications. Their analytical performance evaluation usually relies on the truncated union bound, which provides a low error rate approximation based on the minimum distance term only. In fact, the complete weight enumerator of most product codes remains unknown. In this paper, concatenated representations are introduced and a...

There are different ways to realize Reed Solomon (RS) codes. While in the storage community, using the generator matrices to implement RS codes is more popular, in the coding theory community the generator polynomials are typically used to realize RS codes. Prominent exceptions include HDFS-RAID, which uses generator polynomial based erasure codes, and extends the Apache Hadoop’s file system. I...

We give constructions of probabilistically checkable proofs (PCPs) of length n · polylog n proving satisfiability of circuits of size n that can verified by querying polylog n bits of the proof. We also give analogous constructions of locally testable codes (LTCs) mapping n information bits to n · polylog n bit long codewords that are testable with polylog n queries. Our constructions rely on n...

In this paper, we apply set partitioning to multi-dimensional signal spaces over GF(q), particularly GFq-l(q) and GFq(q), and show how to construct both multi-level block codes and multi-level trellis codes over GF(q). We present two classes of multi-level (n, k, d) block codes over GF(q) with block length n, number of information symbols k, and minimum ,1-1 • { d } distance d_n >_ d, where n =...

Convolutional stabilizer codes promise to make quantum communication more reliable with attractive online encoding and decoding algorithms. This paper introduces a new approach to convolutional stabilizer codes based on direct limit constructions. Two families of quantum convolutional codes are derived from generalized Reed-Solomon codes and from ReedMuller codes. A Singleton bound for pure con...

We study the minimum distance of concatenated codes with Reed-Solomon outer codes and convolutional inner codes. For suitable combinations of parameters the minimum distance may be lower-bounded by the product of the minimum distances of the inner and outer codes. For a randomized ensemble of concatenated codes we prove a GilbertVarsharnov-type lower bound.

In this paper, we extend our work in [4] which employed Partial Reed-Solomon (PRS) Codes at coding rates near channel capacity on a Binary Erasure Channel (BEC). We demonstrated that an appropriately designed PRS code outperforms the classical Reed-Solomon (RS) code for a performance criterion tailored for realtime applications. This paper extends this analysis for a general BEC with varying ch...

– In 1983, Welch, L.R. and Berlekamp, E.R. introduced a new key equation based on rational interpolation for algebraically decoding Reed-Solomon (RS) codes over finite fields. A key feature of their so-called remainder decoding approach is that the computation of the syndrome vector corresponding to a given received word is not needed. For long codes, this implies significant computational savi...

A link budget for a deep-space optical channel depends in part on the choice of modulation format and error-control coding scheme. This article describes several properties of the channel capacity that lead to an appropriate selection of modulation format, pulse-position modulation (PPM) order, and error-control code rate. It also describes performance limits when additional constraints—such as...

Error-correcting codes constitute one of the key ingredients in achieving the high degree of reliability required in modern data transmission and storage systems. This book introduces the reader to the theoretical foundations of error-correcting codes, with an emphasis on Reed–Solomon codes and their derivative codes. After reviewing linear codes and finite fields, the author describes Reed–Sol...