coding scheme design

This work addresses the underappreciated role of algebra in decoding algorithms for error-correcting codes. While algebraic structures underpin code construction, their systematic influence on decoding design—particularly regarding complexity, error-correction capability, and implementation efficiency—remains inadequately characterized. To bridge this gap, we develop a unified “algebra-driven algorithm” framework, integrating finite field theory, polynomial algebra, linear algebra, and algebraic geometry. We rigorously analyze how algebraic properties govern decoding mechanisms across classical and modern codes—including Reed–Solomon and LDPC codes—with emphasis on interpolation-based, syndrome-based, and algebraic soft-decision decoding. Our contribution lies in establishing precise structural links between algebraic invariants (e.g., degree bounds, nullspace structure, curve genus) and algorithmic performance metrics. The framework yields scalable theoretical principles and practical design guidelines for high-reliability, low-latency coding systems. (149 words)

This work addresses two key limitations in structured code design: weak parameter customization and fragmented construction frameworks. We propose a unified construction framework based on unit-derived schemes from Hadamard matrices, enabling systematic generation of linear block codes, classical convolutional codes, and quantum convolutional codes. Integrating Hadamard matrix algebra, coding theory, and quantum error-correction code design, our method supports on-demand specification of code length, rate, and type. For the first time, it achieves joint construction of self-dual codes, dual-containing codes, linear complementary dual (LCD) codes, and both stabilizer and non-stabilizer quantum error-correcting codes. We rigorously derive lower bounds on minimum distance to guarantee error-correction performance. The framework overcomes structural constraints inherent in classical algebraic coding, establishing a general paradigm for parameter-controllable, theoretically verifiable, and application-scalable code design.

This work addresses the challenge of designing high-rate, low-overhead CSS quantum error-correcting codes. We introduce *dihedral quantum codes*—a novel family of CSS codes constructed via the lifted product—and present the first systematic framework for short-length instances. Leveraging the group algebra structure and representation theory of the dihedral group, we derive an explicit analytical formula for the code dimension in terms of two classical constituent codes and rigorously establish a nontrivial lower bound on the quantum distance. Beyond theoretical development, we provide concrete constructions that empirically validate the explicit trade-off between dimension (i.e., encoding rate) and distance (i.e., error-correction capability). Our results unify structural insights from group theory and coding theory, yielding a new design paradigm for practical quantum codes that simultaneously achieve high rate and robust fault tolerance.

This work systematically examines the structures of classical and quantum error-correcting codes in storage and communication, elucidating their deep connections to mathematical and physical objects such as sphere packings, lattices, combinatorial designs, group theory, and quantum phases of matter. The project pioneers a handbook-style integration of the error-correction knowledge base, employing a taxonomic framework that unifies information theory, algebraic coding theory, and interdisciplinary mapping techniques. Codes are organized structurally according to symbol types, enabling coherent classification and cross-referencing. The resulting resource not only serves as a rigorous and comprehensive reference but also empowers researchers to trace interrelationships among codes and inspire novel discoveries, thereby addressing a critical gap in systematic synthesis and cross-domain linkage within the field.

This work addresses the design of error-correcting codes for multiset deletion errors over channels where symbol order is entirely lost. By integrating algebraic and geometric techniques, the authors establish—for the first time—an exact generating function for multiset deletion balls and provide a global characterization of their extremal centers. They uncover an intrinsic connection between the maximum deletion ball volume and ideal difference sets. Leveraging constructions based on polynomial Sidon sets and differential vector representations, they propose $t$-deletion-correcting codes with redundancy $t + O(1)$. Furthermore, they derive precise formulas for deletion ball size and average volume, and establish both a Gilbert–Varshamov-type lower bound and a sphere-packing upper bound that asymptotically match.

This work addresses the construction of quantum LDPC base matrices satisfying regularity, CSS orthogonality, and the absence of 4-cycles of the same type by proposing a finite-field-based dual-branch multiplicative coset method. The design is decomposed into two stages: base matrix construction and cyclic lifting. The former explicitly encodes degree distribution and girth constraints via quotient coset conditions, while the latter employs algebraic randomization to establish edge connections. The proposed framework accommodates various $(J,L)$ degree distributions, offering both flexibility and structural rigor. Experimentally, a CSS code with parameters $[[10240,4108,10\leq d\leq32]]$ is successfully constructed, achieving a frame error rate of $1.0\times10^{-7}$ at a physical error rate of $p=0.058$, exhibiting a Tanner graph girth of at least 8, and excluding weight-16 non-degenerate logical error support orbits.

This study investigates the structure of X- and Z-type parity-check matrices in lifted product quantum LDPC codes and their impact on decoding performance and error floor behavior. By employing graph-theoretic analysis and Tanner graph modeling, the work rigorously establishes—for the first time—that the Tanner graphs associated with \(H_X\) and \(H_Z\) are isomorphic, and derives necessary and sufficient conditions for their connectivity. Building on absorbing set theory, the authors further derive upper and lower bounds on the size of minimal absorbing sets. These results uncover the key combinatorial structures governing error floor phenomena, thereby providing a theoretical foundation for understanding and optimizing the decoding performance of lifted product codes.

This study addresses the limitation of classical perfect and MDS codes, which cannot simultaneously protect data and their function values with distinct error-correction capabilities. The authors establish, for the first time, a graph-theoretic existence framework for strict functional error-correcting codes, recasting linear code construction as a subcode generation problem. Two novel approaches are proposed: a reverse construction method based on weight distribution constraints that reduces the number of minimum-weight codewords, and a construction leveraging narrow-sense BCH codes with designed distance three. By integrating α-distance graphs, Cayley graph isomorphism analysis, and a reverse application of Simonis’s results, the work successfully constructs multiple explicit code families. These constructions not only confirm the existence of such codes but also transcend classical coding-theoretic boundaries, significantly advancing the theoretical foundations of functional error correction.