Moshe Schwartz

Gabriel Sac Himelfarb, Moshe Schwartz

We define and study burst-covering codes. We provide some general bounds connecting the parameters of a code with its burst-covering radius. We then provide stronger bounds on the burst-covering radius of cyclic codes, by employing linear-feedback shift-register (LFSR) sequences. For the case of BCH codes we prove a new bound on pattern frequencies in LFSR sequences, which is of independent interest. Using this tool, we can bound the burst-covering radius of binary primitive BCH codes and Melas codes. We then present an efficient burst-covering algorithm for cyclic codes. Finally, we present a bound on the critical exponent of cyclic codes based on the burst-covering radius.

Wenjun Yu, Moshe Schwartz

A single coloring channel is defined by a subset of letters it allows to pass through, while deleting all others. A sequence of coloring channels provides multiple views of the same transmitted letter sequence, forming a type of sequence-reconstruction problem useful for protein identification and information storage at the molecular level. We provide exact capacities of several sequences of coloring channels: uniform sunflowers, two arbitrary intersecting sets, and paths. We also show how this capacity depends solely on a related graph we define, called the pairs graph. Using this equivalence, we prove lower and upper bounds on the capacity, and a tailored bound for a coloring-channel sequence forming a cycle. In particular, for an alphabet of size $4$, these results give the exact capacity of all coloring-channel sequences except for a cycle of length $4$, for which we only provide bounds.

Farzad Farnoud, Moshe Schwartz, Jehoshua Bruck

It is known that the majority of the human genome consists of repeated sequences. Furthermore, it is believed that a significant part of the rest of the genome also originated from repeated sequences and has mutated to its current form. In this paper, we investigate the possibility of constructing an exponentially large number of sequences from a short initial sequence and simple replication rules, including those resembling genomic replication processes. In other words, our goal is to find out the capacity, or the expressive power, of these string-replication systems. Our results include exact capacities, and bounds on the capacities, of four fundamental string-replication systems.