MinT - Best Known (37−31, 37, s)-Nets in Base 128

Best Known (37−31, 37, s)-Nets in Base 128

(37−31, 37, 216)-Net over F₁₂₈ — Constructive and digital

Digital (6, 37, 216)-net over F₁₂₈, using

t-expansion [i] based on digital (5, 37, 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]

(37−31, 37, 243)-Net over F₁₂₈ — Digital

Digital (6, 37, 243)-net over F₁₂₈, using

net from sequence [i] based on digital (6, 242)-sequence over F₁₂₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 6 and N(F) ≥ 243, using
  - a curve from the manYPoints database [i]

(37−31, 37, 258)-Net in Base 128 — Constructive

(6, 37, 258)-net in base 128, using

3 times m-reduction [i] based on (6, 40, 258)-net in base 128, using
- base change [i] based on digital (1, 35, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(37−31, 37, 289)-Net in Base 128

(6, 37, 289)-net in base 128, using

3 times m-reduction [i] based on (6, 40, 289)-net in base 128, using
- base change [i] based on digital (1, 35, 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]

(37−31, 37, 5764)-Net in Base 128 — Upper bound on s

There is no (6, 37, 5765)-net in base 128, because

1 times m-reduction [i] would yield (6, 36, 5765)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 7254 080195 479073 555184 260837 240148 810408 487907 751901 097000 936931 750536 155776 > 128³⁶ [i]