MinT - Best Known (5, 5+15, s)-Nets in Base 128

Best Known (5, 5+15, s)-Nets in Base 128

(5, 5+15, 216)-Net over F₁₂₈ — Constructive and digital

Digital (5, 20, 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]

(5, 5+15, 227)-Net over F₁₂₈ — Digital

Digital (5, 20, 227)-net over F₁₂₈, using

net from sequence [i] based on digital (5, 226)-sequence over F₁₂₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 227, using
  - a curve from the manYPoints database [i]

(5, 5+15, 259)-Net in Base 128 — Constructive

(5, 20, 259)-net in base 128, using

4 times m-reduction [i] based on (5, 24, 259)-net in base 128, using
- base change [i] based on digital (2, 21, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(5, 5+15, 321)-Net in Base 128

(5, 20, 321)-net in base 128, using

4 times m-reduction [i] based on (5, 24, 321)-net in base 128, using
- base change [i] based on digital (2, 21, 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]

(5, 5+15, 13950)-Net in Base 128 — Upper bound on s

There is no (5, 20, 13951)-net in base 128, because

1 times m-reduction [i] would yield (5, 19, 13951)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 10891 771335 966201 441100 341698 949083 092232 > 128¹⁹ [i]