MinT - Best Known (59−36, 59, s)-Nets in Base 128

Best Known (59−36, 59, s)-Nets in Base 128

(59−36, 59, 345)-Net over F₁₂₈ — Constructive and digital

Digital (23, 59, 345)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 18, 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 (5, 41, 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]

(59−36, 59, 405)-Net over F₁₂₈ — Digital

Digital (23, 59, 405)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(59−36, 59, 513)-Net in Base 128

(23, 59, 513)-net in base 128, using

t-expansion [i] based on (17, 59, 513)-net in base 128, using
- 13 times m-reduction [i] based on (17, 72, 513)-net in base 128, using
  - base change [i] based on digital (8, 63, 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]

(59−36, 59, 480047)-Net in Base 128 — Upper bound on s

There is no (23, 59, 480048)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 21153 909618 929235 526649 753709 694473 324830 098334 894695 403727 335531 824417 399690 765791 624807 467343 774768 962548 096616 575566 504746 > 128⁵⁹ [i]