MinT - Best Known (48−41, 48, s)-Nets in Base 128

Best Known (48−41, 48, s)-Nets in Base 128

(48−41, 48, 216)-Net over F₁₂₈ — Constructive and digital

Digital (7, 48, 216)-net over F₁₂₈, using

t-expansion [i] based on digital (5, 48, 216)-net over F₁₂₈, using
- net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
    - a function field by SÃ©mirat [i]

(48−41, 48, 258)-Net in Base 128 — Constructive

(7, 48, 258)-net in base 128, using

base change [i] based on digital (1, 42, 258)-net over F₂₅₆, using
- net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(48−41, 48, 262)-Net over F₁₂₈ — Digital

Digital (7, 48, 262)-net over F₁₂₈, using

net from sequence [i] based on digital (7, 261)-sequence over F₁₂₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 7 and N(F) ≥ 262, using
  - a curve from the manYPoints database [i]

(48−41, 48, 289)-Net in Base 128

(7, 48, 289)-net in base 128, using

base change [i] based on digital (1, 42, 289)-net over F₂₅₆, using
- net from sequence [i] based on digital (1, 288)-sequence over F₂₅₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289, using
    - sharp bound on the number of rational points on curves with genus 1 [i]

(48−41, 48, 5844)-Net in Base 128 — Upper bound on s

There is no (7, 48, 5845)-net in base 128, because

1 times m-reduction [i] would yield (7, 47, 5845)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 1096 415214 012946 355899 726479 095757 356385 886982 245461 514163 460952 870325 083830 995398 149875 886678 704693 > 128⁴⁷ [i]