MinT - Best Known (4, 16, s)-Nets in Base 64

Best Known (4, 16, s)-Nets in Base 64

(4, 16, 104)-Net over F₆₄ — Constructive and digital

Digital (4, 16, 104)-net over F₆₄, using

t-expansion [i] based on digital (3, 16, 104)-net over F₆₄, using
- net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
    - a function field by SÃ©mirat [i]

(4, 16, 129)-Net over F₆₄ — Digital

Digital (4, 16, 129)-net over F₆₄, using

net from sequence [i] based on digital (4, 128)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 4 and N(F) ≥ 129, using
  - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₆₄ [i]

(4, 16, 257)-Net in Base 64 — Constructive

(4, 16, 257)-net in base 64, using

base change [i] based on digital (0, 12, 257)-net over F₂₅₆, using
- net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
  - generalized Faure sequence [i]
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
    - the rational function field F₂₅₆(x) [i]
  - Niederreiter sequence [i]

(4, 16, 3111)-Net in Base 64 — Upper bound on s

There is no (4, 16, 3112)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 79272 040126 271674 849676 214175 > 64¹⁶ [i]