Applications of Derandomization Theory in Coding

Randomized techniques play a fundamental role in theoretical computer science and discrete mathematics, in particular for the design of efficient algorithms and construction of combinatorial objects. The basic goal in derandomization theory is to eliminate or reduce the need for randomness in such randomized constructions. In this thesis, we explore some applications of the fundamental notions in derandomization theory to problems outside the core of theoretical computer science, and in particular, certain problems related to coding theory. First, we consider the wiretap channel problem which involves a communication system in which an intruder can eavesdrop a limited portion of the transmissions, and construct efficient and information-theoretically optimal communication protocols for this model. Then we consider the combinatorial group testing problem. In this classical problem, one aims to determine a set of defective items within a large population by asking a number of queries, where each query reveals whether a defective item is present within a specified group of items. We use randomness condensers to explicitly construct optimal, or nearly optimal, group testing schemes for a setting where the query outcomes can be highly unreliable, as well as the threshold model where a query returns positive if the number of defectives pass a certain threshold. Finally, we design ensembles of error-correcting codes that achieve the information-theoretic capacity of a large class of communication channels, and then use the obtained ensembles for construction of explicit capacity achieving codes. [This is a shortened version of the actual abstract in the thesis.]

💡 Research Summary

This dissertation explores how fundamental tools from derandomization theory—pseudorandom generators, randomness extractors, and randomness condensers—can be employed to solve concrete problems in coding theory without relying on large amounts of true randomness. The work is organized into four main parts, each addressing a distinct application.

The first part tackles the wiretap channel, a model where an eavesdropper can observe a limited fraction of transmitted symbols. Classical solutions rely on random coding arguments that are difficult to implement. By leveraging symbol‑fixing extractors and restricted affine extractors, the author constructs explicit linear‑code‑based schemes that achieve information‑theoretic secrecy with only a small secret key. The constructions are efficient: encoding and decoding can be performed in near‑linear time, and the resulting protocols meet the optimal secrecy capacity. A random‑walk based wiretap protocol is also presented, illustrating how the extractor framework yields practical, low‑overhead secure communication.

The second part addresses combinatorial group testing, especially in noisy environments and in the threshold model where a test returns positive only if the number of defectives exceeds a given threshold. Traditional designs either assume error‑free responses or rely on probabilistic constructions. Here, the author exploits the almost‑injectivity property of lossless condensers to build explicit test matrices that are robust to high error rates and that achieve the information‑theoretic lower bound on the number of tests up to constant factors. The condenser‑based matrices have a simple algebraic structure, which translates into reduced storage and fast decoding algorithms, making them attractive for practical large‑scale screening applications.

The third part focuses on constructing ensembles of error‑correcting codes that achieve the Shannon capacity of a broad class of memoryless channels. By combining extractors (to amplify weak randomness in the channel output) with condensers (to concentrate entropy), the author designs explicit code families whose rate approaches channel capacity while guaranteeing a prescribed error exponent. Unlike classical random‑code arguments that prove existence without construction, the presented ensembles are explicitly describable and admit efficient encoding/decoding procedures. The analysis quantifies how quickly the gap to capacity shrinks as block length grows, providing a concrete pathway from theory to implementation.

The final chapter revisits the Gilbert‑Varshamov bound, which asserts the existence of codes with relative distance close to the optimal trade‑off between rate and distance. Building on the Nisan‑Wigderson paradigm, the dissertation extends the framework to produce a small explicit collection of codes that, under plausible computational hardness assumptions (e.g., hardness of certain Boolean functions against small circuits), achieve the GV bound for most codes in the collection. This result bridges the gap between non‑constructive existence proofs and practical code design, offering explicit constructions that retain the asymptotic optimality of random codes while being amenable to algorithmic implementation.

Overall, the thesis demonstrates that derandomization tools are not merely of theoretical interest; they can be systematically transformed into concrete, efficient, and provably optimal constructions for core coding‑theoretic problems. By reducing or eliminating the need for true randomness, the work paves the way for more reliable, secure, and implementable communication systems.