Uncorrectable Errors of Weight Half the Minimum Distance for Binary Linear Codes

A lower bound on the number of uncorrectable errors of weight half the minimum distance is derived for binary linear codes satisfying some condition. The condition is satisfied by some primitive BCH codes, extended primitive BCH codes, Reed-Muller codes, and random linear codes. The bound asymptotically coincides with the corresponding upper bound for Reed-Muller codes and random linear codes. By generalizing the idea of the lower bound, a lower bound on the number of uncorrectable errors for weights larger than half the minimum distance is also obtained, but the generalized lower bound is weak for large weights. The monotone error structure and its related notion larger half and trial set, which are introduced by Helleseth, Kl{\o}ve, and Levenshtein, are mainly used to derive the bounds.Comment: 5 pages, to appear in ISIT 200

Capacity-Achieving Coding Mechanisms: Spatial Coupling and Group Symmetries

The broad theme of this work is in constructing optimal transmission mechanisms for a wide variety of communication systems. In particular, this dissertation provides a proof of threshold saturation for spatially-coupled codes, low-complexity capacity-achieving coding schemes for side-information problems, a proof that Reed-Muller and primitive narrow-sense BCH codes achieve capacity on erasure channels, and a mathematical framework to design delay sensitive communication systems. Spatially-coupled codes are a class of codes on graphs that are shown to achieve capacity universally over binary symmetric memoryless channels (BMS) under belief-propagation decoder. The underlying phenomenon behind spatial coupling, known as “threshold saturation via spatial coupling”, turns out to be general and this technique has been applied to a wide variety of systems. In this work, a proof of the threshold saturation phenomenon is provided for irregular low-density parity-check (LDPC) and low-density generator-matrix (LDGM) ensembles on BMS channels. This proof is far simpler than published alternative proofs and it remains as the only technique to handle irregular and LDGM codes. Also, low-complexity capacity-achieving codes are constructed for three coding problems via spatial coupling: 1) rate distortion with side-information, 2) channel coding with side-information, and 3) write-once memory system. All these schemes are based on spatially coupling compound LDGM/LDPC ensembles. Reed-Muller and Bose-Chaudhuri-Hocquengham (BCH) are well-known algebraic codes introduced more than 50 years ago. While these codes are studied extensively in the literature it wasn’t known whether these codes achieve capacity. This work introduces a technique to show that Reed-Muller and primitive narrow-sense BCH codes achieve capacity on erasure channels under maximum a posteriori (MAP) decoding. Instead of relying on the weight enumerators or other precise details of these codes, this technique requires that these codes have highly symmetric permutation groups. In fact, any sequence of linear codes with increasing blocklengths whose rates converge to a number between 0 and 1, and whose permutation groups are doubly transitive achieve capacity on erasure channels under bit-MAP decoding. This pro-vides a rare example in information theory where symmetry alone is sufficient to achieve capacity. While the channel capacity provides a useful benchmark for practical design, communication systems of the day also demand small latency and other link layer metrics. Such delay sensitive communication systems are studied in this work, where a mathematical framework is developed to provide insights into the optimal design of these systems