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Main discovery. For fixed finite field size q, and for diameters 4 ≤ d ≤ n−1 (excluding the binary endpoint q=2, d=n−1), the total number of d‑anticodes in F_q^n satisfies |𝒜_q(n,d)| = 2^{(1+o(1)) m_q(n,d)}. A d‑anticode is any subset whose pairwise Hamming distances are at most d (equivalently, a set of diameter at most d). (This equality is asymptotic as n→∞.)
Method and key steps. The proof uses a Helly‑type theorem for Hamming balls. Every inclusion‑maximal d‑anticode can be written as the intersection of Hamming balls centered at at most 2d+1 of its own points. In other words, each inclusion‑maximal anticode is determined by ≤2d+1 points. This permits encoding each maximal anticode by an ordered list of 2d+1 points. Counting such lists gives an upper bound on the number of maximal anticodes and thus controls the total number of anticodes. A simple lower bound holds because every subset of a maximum d‑anticode is itself a d‑anticode, so |𝒜_q(n,d)| ≥ 2^{m_q(n,d)}. Combining the lower bound with the Helly‑based upper bound yields the asymptotic equality above for the stated range of d and q.
Concrete results and exceptions. - For 4 ≤ d ≤ n−1 (fixed q, n→∞), except when q=2 and d=n−1, |𝒜_q(n,d)| = 2^{(1+o(1)) m_q(n,d)}. - For small fixed diameters d∈{1,2,3} and fixed q, the total count grows only exponentially in n on a logarithmic scale: |𝒜_q(n,d)| = 2^{Θ_q(n)}. - At the binary endpoint d = n−1 there is an explicit exception: the vectors of F_2^n split into 2^{n−1} complementary pairs, each pair offers three choices (choose neither element or exactly one), so |𝒜_2(n,n−1)| = 3^{2^{n−1}}. The maximum anticode size is m_2(n,n−1)=2^{n−1}. Hence log_2 |𝒜_2(n,n−1)| = (log_2 3)·2^{n−1} = (log_2 3)·m_2(n,n−1), and equivalently |𝒜_2(n,n−1)| = 2^{(log_2 3) m_2(n,n−1)}, which differs from the (1+o(1)) exponent in the general law.
Linear anticodes (subspaces of diameter ≤ d). Counting linear d‑anticodes behaves differently and was treated separately. The main statements are: 1) If d is fixed, there exists an integer σ_q(d) with d ≤ σ_q(d) ≤ ⌊qd/(q−1)⌋ such that |L_q(n,d)| = Θ_{q,d}(n^{σ_q(d)}). In particular, if q > d+1 then σ_q(d)=d and |L_q(n,d)| = Θ_{q,d}(n^d). 2) If d→∞ but d=o(n), then |L_q(n,d)| = q^{Θ_q(d^2 + d log_q(n/d))}. 3) If d/n → α ∈ (0,1) and γ = min{1, qα/(q−1)}, then q^{(α^2/4+o(1)) n^2} ≤ |L_q(n,d)| ≤ q^{(γ^2/4+o(1)) n^2}. 4) If n−d = o(n) (near‑full diameter), then |L_q(n,d)| = q^{(1/4+o(1)) n^2}.
A refined binary construction. For q=2 and d = ⌊αn⌋ with 0<α<1, a random‑generator construction gives an improved lower bound: |L_2(n,d)| ≥ 2^{(φ(α)+o(1)) n^2}, where φ(α) is obtained by optimizing certain parameters. This strictly improves the α^2/4 exponent for every 0<α<1.
Why the difference between general and linear counts. Asymptotically (on the logarithmic scale) the general anticode count for 4≤d≤n−1 (excluding the binary endpoint) is governed by the number of subsets of a single maximum anticode. Inclusion‑maximal anticodes are asymptotically negligible in that range. Imposing linearity changes the asymptotic growth. The linear case is organized by the support size of a subspace and requires separate combinatorial analysis.
Caveats and limitations. - The Helly approach was used because the authors report they lack a uniform stability result for q≥3 across the full range of diameters; that absence prevented a container‑method approach in some regimes. - A basic support bound holds for full‑support linear anticodes: if a linear code C ≤ F_q^s has diameter ≤ d then its support size satisfies s ≤ qd/(q−1). This bound is asymptotically tight as d→∞; explicit constructions (using q‑ary simplex codes plus small blocks) achieve support size (q/(q−1)+o(1)) d. - The present results fix q while n→∞. Determining asymptotics when q = q(n) and d = d(n) grow simultaneously is left as an open problem.
All numerical expressions, comparison conditions, uncertainties, and limitations above are stated as in the cited results.