MinT - Best Known (29−23, 29, s)-Nets in Base 128

Best Known (29−23, 29, s)-Nets in Base 128

(29−23, 29, 216)-Net over F₁₂₈ — Constructive and digital

Digital (6, 29, 216)-net over F₁₂₈, using

t-expansion [i] based on digital (5, 29, 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]

(29−23, 29, 243)-Net over F₁₂₈ — Digital

Digital (6, 29, 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]

(29−23, 29, 259)-Net in Base 128 — Constructive

(6, 29, 259)-net in base 128, using

3 times m-reduction [i] based on (6, 32, 259)-net in base 128, using
- base change [i] based on digital (2, 28, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(29−23, 29, 321)-Net in Base 128

(6, 29, 321)-net in base 128, using

3 times m-reduction [i] based on (6, 32, 321)-net in base 128, using
- base change [i] based on digital (2, 28, 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]

(29−23, 29, 8928)-Net in Base 128 — Upper bound on s

There is no (6, 29, 8929)-net in base 128, because

1 times m-reduction [i] would yield (6, 28, 8929)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 100516 331787 684789 951654 831829 982170 293893 697683 648691 353056 > 128²⁸ [i]