MinT - Best Known (25−17, 25, s)-Nets in Base 128

Best Known (25−17, 25, s)-Nets in Base 128

(25−17, 25, 258)-Net over F₁₂₈ — Constructive and digital

Digital (8, 25, 258)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 8, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
      - the rational function field F₁₂₈(x) [i]
    - Niederreiter sequence [i]
2. digital (0, 17, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈ (see above)

(25−17, 25, 261)-Net in Base 128 — Constructive

(8, 25, 261)-net in base 128, using

7 times m-reduction [i] based on (8, 32, 261)-net in base 128, using
- base change [i] based on digital (4, 28, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(25−17, 25, 276)-Net over F₁₂₈ — Digital

Digital (8, 25, 276)-net over F₁₂₈, using

net from sequence [i] based on digital (8, 275)-sequence over F₁₂₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 8 and N(F) ≥ 276, using
  - a curve from the manYPoints database [i]

(25−17, 25, 321)-Net in Base 128

(8, 25, 321)-net in base 128, using

23 times m-reduction [i] based on (8, 48, 321)-net in base 128, using
- base change [i] based on digital (2, 42, 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]

(25−17, 25, 62157)-Net in Base 128 — Upper bound on s

There is no (8, 25, 62158)-net in base 128, because

1 times m-reduction [i] would yield (8, 24, 62158)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 374 183513 322844 473604 174551 029560 717892 968404 912095 > 128²⁴ [i]