Code Invariants in Classical and Quantum Error Correction

Error-correcting codes are mathematical structures that keep communication, data storage, and computation reliable when noise and hardware faults are unavoidable. They already underpin everyday technology, from cell phones and the internet to deep-space communication, and they are now central to quantum information science. Quantum computers are extremely sensitive to errors, and quantum error correction is one of the key requirements for building reliable large-scale quantum devices. A core difficulty is knowing, in a rigorous way, which codes are good, how to compare them, and how to build new ones with provable quality. This project develops a unified mathematical framework that addresses these questions for both classical and quantum codes, revealing shared structure beneath families long studied with separate tools and establishing a common language for carrying techniques from classical coding theory into quantum error correction. This perspective is expected to help clarify the limits of quantum error correction and guide the construction of stronger codes. The project will also train undergraduate and graduate students through research and computational experimentation, and will reach the wider public, strengthening the talent pipeline in mathematics and quantum technology, a national priority recognized in the National Quantum Initiative Act. This project develops an invariant theory for error-correcting codes and applies it across the Hamming, rank, tensor, and symplectic settings, extending invariant-theoretic tools that have largely been developed separately for each metric. The central insight is that generalized weights, weight distributions, and their duality theory can be derived by studying the interaction of codes with structured algebraic and combinatorial objects, namely anticodes. This makes it possible to recover, extend, and unify classical and quantum results within a common framework. The first direction applies this setting to symplectic codes, an algebraic framework that also models stabilizer quantum codes, producing new invariants, bounds, and MacWilliams- and Wei-type duality theorems, and recasting quantum information-theoretic principles, such as the Cleaning Lemma and complementary recovery, in combinatorial and algebraic terms. The second direction extends invariant theory from matrix to higher-order tensor rank-metric codes, classifying anticode families that capture distinct notions of support, establishing associated duality results, and constructing extremal and optimal families relevant to network coding and post-quantum cryptography. The third direction studies a further invariant arising from this framework, the tensor rank of structured rank-metric codes, such as one-dimensional Gabidulin codes, linking coding theory to algebraic complexity and the long-standing problem of efficient multiplication in finite fields. Since codes of low tensor rank admit more efficient storage and encoding, this direction also connects fundamental theory to the design of more practical communication systems. Together, these directions integrate methods from algebra, combinatorics, algebraic geometry, finite geometry, matroid theory, and complexity theory to advance the mathematical foundations of error correction. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria. NSF Award ID: 2554002 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Giuseppe Cotardo | Institution: Virginia Polytechnic Institute and State University, BLACKSBURG, VA | Award Amount: $200,000 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2554002 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2554002.html