MinT - Best Known (22−15, 22, s)-Nets in Base 128

Best Known (22−15, 22, s)-Nets in Base 128

(22−15, 22, 258)-Net over F₁₂₈ — Constructive and digital

Digital (7, 22, 258)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 7, 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, 15, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈ (see above)

(22−15, 22, 261)-Net in Base 128 — Constructive

(7, 22, 261)-net in base 128, using

2 times m-reduction [i] based on (7, 24, 261)-net in base 128, using
- base change [i] based on digital (4, 21, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(22−15, 22, 262)-Net over F₁₂₈ — Digital

Digital (7, 22, 262)-net over F₁₂₈, using

net from sequence [i] based on digital (7, 261)-sequence over F₁₂₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 7 and N(F) ≥ 262, using
  - a curve from the manYPoints database [i]

(22−15, 22, 321)-Net in Base 128

(7, 22, 321)-net in base 128, using

18 times m-reduction [i] based on (7, 40, 321)-net in base 128, using
- base change [i] based on digital (2, 35, 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]

(22−15, 22, 55811)-Net in Base 128 — Upper bound on s

There is no (7, 22, 55812)-net in base 128, because

1 times m-reduction [i] would yield (7, 21, 55812)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 178 424715 841955 600369 356034 614304 655712 327552 > 128²¹ [i]