Even-Order Groups Exhibit Fixed Arithmetic Limits, Unlike Their Odd-Order Counterparts

Scientists investigating restricted sumsets in finite abelian groups have identified critical numerical thresholds governing the transition from sparse to saturated additive structures. Bocong Chen from the School of Mathematics, South China University of Technology, and Jing Huang from the School of Mathematics and Information Science, Guangzhou University, present a comprehensive classification of these critical numbers, establishing exact values for even-order groups and providing rigorous bounds for odd-order groups. This collaborative work reveals a fundamental structural dichotomy dictated by group parity, demonstrating rigidity in even-order groups where a fixed density barrier prevails, and collapse in odd-order groups where this barrier vanishes, leading to significantly lower densities. These findings generalise previous results on cyclic groups and offer a definitive structural theory, crucially resolving a long-standing conjecture by Han and Ren concerning the construction of minimum distance separable (MDS) codes from elliptic curve rational points.

Scientists have uncovered a fundamental principle governing the behaviour of restricted sumsets within finite abelian groups, revealing a surprising dichotomy dictated by whether the group’s order is even or odd. This work establishes, for the first time, precise classifications of ‘critical numbers’, thresholds determining when a set’s sums can fully ‘cover’ the group. For groups with an even number of elements, a rigid arithmetic barrier emerges at a density of 1/2, fixing the critical number at |G|/2+1 irrespective of the group’s internal complexity. In stark contrast, research demonstrates that odd-order groups lack this barrier, allowing the critical threshold to plummet to significantly lower densities, influenced by the group’s smallest prime divisor or index-5 obstructions. These findings not only unify and extend previous results limited to cyclic groups, but also provide a definitive structural theory explaining the transition from sparse to saturated sumsets. The study’s implications extend beyond pure mathematics, offering a resolution to a longstanding conjecture in algebraic coding theory. By translating the observed additive rigidity into a geometric constraint, researchers prove a tight bound on the size of subsets of rational points on elliptic curves that generate maximum distance separable (MDS) codes, a crucial component in error-correcting codes used in data transmission and storage. The research rigorously defines the k-critical number as the minimum size a subset must reach to guarantee its k-fold restricted sumset equals the entire group. Theorem A establishes a global sufficiency condition, demonstrating that any subset exceeding a specific size, dependent on the group’s order, will generate the complete group for all admissible lengths k. Theorem B then delves into precise bounds and exact values for these critical numbers, confirming the parity-dependent behaviour. Specifically, even-order groups exhibit a universal rigidity, while odd-order groups display a collapse of the critical density, governed by index-5 obstructions or the smallest prime divisor. This work provides a powerful new framework for understanding additive structures in finite abelian groups and their connections to practical applications in information theory. For groups of even order, research demonstrates a universal rigidity wherein the index-2 subgroup establishes an immutable arithmetic barrier at a density of 1/2, fixing the critical number at |G|/2 + 1 irrespective of the group’s internal composition. This means that any subset of the group must exceed half its total size to guarantee complete coverage of the group through restricted sumsets. The study rigorously proves this threshold, extending prior work on cyclic groups to encompass all finite abelian groups with an even number of elements. In stark contrast, the work reveals that for groups of odd order, this density 1/2 barrier disappears, causing the critical threshold to fall to significantly lower densities. These lower densities are bounded by index-5 obstructions or the smallest prime divisor of the group’s order. This collapse signifies that smaller subsets are sufficient to generate the entire group via restricted sumsets when the group’s order is odd. As a specific application, the research resolves a conjecture by Han and Ren in algebraic coding theory. The team proves that for all sufficiently large q, any subset of rational points on an elliptic curve E/Fq generating a maximum distance separable (MDS) code must satisfy the tight bound |P| ≤ |E(Fq)|/2. This bound establishes a precise limit on the size of the code’s generating subset relative to the total number of rational points on the curve. Theorem A establishes global sufficiency of the 1/2-density barrier, requiring that for groups of order g, and assuming g ≥ 624 when the smallest prime divisor is 2, g ≥ 3705 when the smallest prime divisor is 3, g ≥ 6175 when the smallest prime divisor is 5, and g ≥ 46319 when the smallest prime divisor is greater than or equal to 7, any subset A with size exceeding g/2 will generate the entire group through restricted sumsets of lengths between 3 and |A| -3. Theorem B then details the precise critical numbers μk(G), confirming the parity-dependent dichotomy and generalizing Bajnok’s cyclic result to arbitrary abelian groups. A meticulous analysis of restricted sumsets underpinned this work, employing a density-based approach to classify critical numbers. The study commenced by establishing precise conditions for when a subset A of a group G generates its entire sumset Γk(A), the set of all possible sums of k distinct elements from A. To delineate this transition from sparsity to saturation, researchers focused on the k-critical number μk(G), defined as the minimum size of A required to guarantee Γk(A) equals G. if the size of A exceeds g/2 (where g is the order of G) and satisfies specific size constraints, g ≥624 |G| + 1846 if p(G) = 2, g ≥3705 if p(G) = 3, g ≥6175 if p(G) = 5, and g ≥46319 if p(G) ≥7, then Γk(A) is guaranteed to equal G for all admissible lengths k. In contrast, the investigation of odd-order groups revealed a markedly different behaviour. Absent the index-2 obstruction, the critical density collapses, becoming governed by smaller index obstructions or the smallest prime divisor of the group’s order. Researchers derived upper bounds for μk(G) in odd-order groups, defining a density constant c(g) dependent on the group’s structure, specifically, c(g) = (2/5 if 5 | g, 5/13 if 5 ∤g), and establishing bounds based on this constant and the group order. This approach allowed for a precise quantification of the parity-dependent dichotomy observed in the behaviour of restricted sumsets. This work delivers a fundamental structural understanding of how sums behave within finite groups, distinguishing sharply between those with even and odd order. For years, establishing definitive limits on how densely a subset of a group can be before its sums begin to ‘saturate’ the entire group felt like a problem yielding only case-by-case solutions. This research transcends that, providing a unifying theory applicable across all abelian groups. The implications extend beyond pure mathematics, notably resolving a longstanding conjecture in algebraic coding theory concerning the maximum length of efficient error-correcting codes derived from elliptic curves. This connection highlights the surprising interplay between abstract algebra and practical data transmission. However, the results for odd-order groups remain expressed as bounds dependent on the group’s smallest prime divisor. Future work will undoubtedly focus on tightening these bounds and exploring whether even finer structural properties govern sumsets in odd-order groups. Moreover, extending these ideas to non-abelian groups represents a natural and ambitious next step. The demonstrated link to coding theory also suggests potential avenues for designing more robust and efficient communication systems, leveraging the newly understood arithmetic constraints.