MinT - Best Known (58−30, 58, s)-Nets in Base 9

Best Known (58−30, 58, s)-Nets in Base 9

(58−30, 58, 78)-Net over F₉ — Constructive and digital

Digital (28, 58, 78)-net over F₉, using

t-expansion [i] based on digital (22, 58, 78)-net over F₉, using
- net from sequence [i] based on digital (22, 77)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
    - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(58−30, 58, 84)-Net in Base 9 — Constructive

(28, 58, 84)-net in base 9, using

2 times m-reduction [i] based on (28, 60, 84)-net in base 9, using
- base change [i] based on digital (8, 40, 84)-net over F₂₇, using
  - net from sequence [i] based on digital (8, 83)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 8 and N(F) ≥ 84, using
      - a function field by SÃ©mirat [i]

(58−30, 58, 113)-Net over F₉ — Digital

Digital (28, 58, 113)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(58−30, 58, 3921)-Net in Base 9 — Upper bound on s

There is no (28, 58, 3922)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 22 242241 270588 710987 815148 894767 085307 684856 239405 836081 > 9⁵⁸ [i]