Asymptotic constant comparison for AECC–profile intersections

Prove that for any integers q, ℓ, and d, if C(H,a) is a Varshamov-type (N, d+1) asymmetric error-correcting code over the prime field Z/pZ with parity-check matrix H chosen so that C(H,a) contains the all-ones vector, then the leading coefficient c(H,S) governing the asymptotic size of the intersection C(H,a) ∩ (n; S) satisfies c(H,S) ≥ c(q,ℓ)/p^d, where c(q,ℓ) is the leading coefficient in the asymptotic expansion of |(n; q,ℓ)| (the number of profile vectors for all q-ary ℓ-grams) and c(H,S) is the leading coefficient defined for the intersection per Corollary 7.2.

Background

The paper studies counts of profile vectors via Ehrhart theory. For S = q^ℓ (all ℓ-grams), the number of profile vectors |(n; S)| has an asymptotic expansion with leading coefficient c(q,ℓ), and when D(S) contains a loop this coefficient is constant. For general constrained S, a similar leading term exists but may be periodic; the authors compute c(q,ℓ) numerically for moderate parameters.

They also construct codes by intersecting a Varshamov AECC C(H,a) (defined by a Vandermonde-type parity-check matrix over Z/pZ) with the set of profile vectors (n; S), and show that the intersection size has an asymptotic expansion with leading coefficient c(H,S). By the pigeonhole principle, one expects an intersection size roughly |(n; S)|/p^d; the conjecture formalizes an asymptotic version of this expectation, asserting that the specific construction achieves at least c(q,ℓ)/p^d in the leading term.