MinT - Best Known (36−22, 36, s)-Nets in Base 128

Best Known (36−22, 36, s)-Nets in Base 128

(36−22, 36, 321)-Net over F₁₂₈ — Constructive and digital

Digital (14, 36, 321)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 11, 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 (3, 25, 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]

(36−22, 36, 353)-Net over F₁₂₈ — Digital

Digital (14, 36, 353)-net over F₁₂₈, using

net from sequence [i] based on digital (14, 352)-sequence over F₁₂₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 14 and N(F) ≥ 353, using
  - a curve from the manYPoints database [i]

(36−22, 36, 513)-Net in Base 128

(14, 36, 513)-net in base 128, using

12 times m-reduction [i] based on (14, 48, 513)-net in base 128, using
- base change [i] based on digital (8, 42, 513)-net over F₂₅₆, using
  - net from sequence [i] based on digital (8, 512)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 8 and N(F) ≥ 513, using
      - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₂₅₆ [i]

(36−22, 36, 304457)-Net in Base 128 — Upper bound on s

There is no (14, 36, 304458)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 7237 129052 936071 240752 934500 495091 095177 678277 754999 070960 859253 394291 228512 > 128³⁶ [i]