MinT - Best Known (16−11, 16, s)-Nets in Base 128

Best Known (16−11, 16, s)-Nets in Base 128

(16−11, 16, 258)-Net over F₁₂₈ — Constructive and digital

Digital (5, 16, 258)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 5, 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, 11, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈ (see above)

(16−11, 16, 260)-Net in Base 128 — Constructive

(5, 16, 260)-net in base 128, using

base change [i] based on digital (3, 14, 260)-net over F₂₅₆, using
- net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(16−11, 16, 321)-Net in Base 128

(5, 16, 321)-net in base 128, using

8 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]

(16−11, 16, 43017)-Net in Base 128 — Upper bound on s

There is no (5, 16, 43018)-net in base 128, because

1 times m-reduction [i] would yield (5, 15, 43018)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 40 568726 505354 609867 326095 452034 > 128¹⁵ [i]