MinT - Best Known (38−31, 38, s)-Nets in Base 128

Best Known (38−31, 38, s)-Nets in Base 128

(38−31, 38, 216)-Net over F₁₂₈ — Constructive and digital

Digital (7, 38, 216)-net over F₁₂₈, using

t-expansion [i] based on digital (5, 38, 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]

(38−31, 38, 259)-Net in Base 128 — Constructive

(7, 38, 259)-net in base 128, using

2 times m-reduction [i] based on (7, 40, 259)-net in base 128, using
- base change [i] based on digital (2, 35, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(38−31, 38, 262)-Net over F₁₂₈ — Digital

Digital (7, 38, 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]

(38−31, 38, 321)-Net in Base 128

(7, 38, 321)-net in base 128, using

2 times m-reduction [i] based on (7, 40, 321)-net in base 128, using
- base change [i] based on digital (2, 35, 321)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 320)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321, using
      - sharp bound on the number of rational points on curves with genus 2 [i]

(38−31, 38, 7968)-Net in Base 128 — Upper bound on s

There is no (7, 38, 7969)-net in base 128, because

1 times m-reduction [i] would yield (7, 37, 7969)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 927745 396147 926503 148753 499168 149140 270946 070306 038191 303715 663341 200680 837280 > 128³⁷ [i]