MinT - Best Known (49−36, 49, s)-Nets in Base 64

Best Known (49−36, 49, s)-Nets in Base 64

(49−36, 49, 177)-Net over F₆₄ — Constructive and digital

Digital (13, 49, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 49, 177)-net over F₆₄, using
- net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
    - a function field by GarcÃa and Quoos [i]

(49−36, 49, 257)-Net in Base 64 — Constructive

(13, 49, 257)-net in base 64, using

3 times m-reduction [i] based on (13, 52, 257)-net in base 64, using
- base change [i] based on digital (0, 39, 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]

(49−36, 49, 257)-Net over F₆₄ — Digital

Digital (13, 49, 257)-net over F₆₄, using

t-expansion [i] based on digital (12, 49, 257)-net over F₆₄, using
- net from sequence [i] based on digital (12, 256)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 12 and N(F) ≥ 257, using
    - K_1,2 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₆₄ [i]

(49−36, 49, 9890)-Net in Base 64 — Upper bound on s

There is no (13, 49, 9891)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 31845 780328 316204 152217 090547 356412 531010 357218 589191 307213 756967 189322 144331 547897 109358 > 64⁴⁹ [i]