MinT - Best Known (1, ∞, s)-Nets in Base 49

Best Known (1, ∞, s)-Nets in Base 49

(1, ∞, 51)-Net over F₄₉ — Constructive and digital

Digital (1, m, 51)-net over F₄₉ for arbitrarily large m, using

net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
- Niederreiter sequence [i]

(1, ∞, 64)-Net over F₄₉ — Digital

Digital (1, m, 64)-net over F₄₉ for arbitrarily large m, using

net from sequence [i] based on digital (1, 63)-sequence over F₄₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 1 and N(F) ≥ 64, using
  - sharp bound on the number of rational points on curves with genus 1 [i]

(1, ∞, 100)-Net in Base 49 — Upper bound on s

There is no (1, m, 101)-net in base 49 for arbitrarily large m, because

m-reduction [i] would yield (1, 99, 101)-net in base 49, but
- extracting embedded OOA [i] would yield OA(49⁹⁹, 101, S₄₉, 98), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 10 461838 291314 357175 018899 611816 813659 819188 550170 233659 950140 084035 125767 424262 251774 382614 909364 050293 065248 252546 314174 063180 343683 591188 150754 267339 816534 637456 120001 / 33 > 49⁹⁹ [i]