MinT - Best Known (19−15, 19, s)-Nets in Base 128

Best Known (19−15, 19, s)-Nets in Base 128

(19−15, 19, 192)-Net over F₁₂₈ — Constructive and digital

Digital (4, 19, 192)-net over F₁₂₈, using

t-expansion [i] based on digital (3, 19, 192)-net over F₁₂₈, using
- net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
    - a function field by SÃ©mirat [i]

(19−15, 19, 215)-Net over F₁₂₈ — Digital

Digital (4, 19, 215)-net over F₁₂₈, using

net from sequence [i] based on digital (4, 214)-sequence over F₁₂₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 4 and N(F) ≥ 215, using
  - a curve from the manYPoints database [i]

(19−15, 19, 258)-Net in Base 128 — Constructive

(4, 19, 258)-net in base 128, using

5 times m-reduction [i] based on (4, 24, 258)-net in base 128, using
- base change [i] based on digital (1, 21, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(19−15, 19, 289)-Net in Base 128

(4, 19, 289)-net in base 128, using

5 times m-reduction [i] based on (4, 24, 289)-net in base 128, using
- base change [i] based on digital (1, 21, 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]

(19−15, 19, 6973)-Net in Base 128 — Upper bound on s

There is no (4, 19, 6974)-net in base 128, because

1 times m-reduction [i] would yield (4, 18, 6974)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 85 094296 481917 126031 219610 612853 853892 > 128¹⁸ [i]