MinT - Best Known (5, 5+21, s)-Nets in Base 128

Best Known (5, 5+21, s)-Nets in Base 128

(5, 5+21, 216)-Net over F₁₂₈ — Constructive and digital

Digital (5, 26, 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+21, 227)-Net over F₁₂₈ — Digital

Digital (5, 26, 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+21, 258)-Net in Base 128 — Constructive

(5, 26, 258)-net in base 128, using

6 times m-reduction [i] based on (5, 32, 258)-net in base 128, using
- base change [i] based on digital (1, 28, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(5, 5+21, 289)-Net in Base 128

(5, 26, 289)-net in base 128, using

6 times m-reduction [i] based on (5, 32, 289)-net in base 128, using
- base change [i] based on digital (1, 28, 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]

(5, 5+21, 6605)-Net in Base 128 — Upper bound on s

There is no (5, 26, 6606)-net in base 128, because

1 times m-reduction [i] would yield (5, 25, 6606)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 47936 746745 715151 843336 052222 213405 141067 030841 593006 > 128²⁵ [i]