MinT - Best Known (14, 53, s)-Nets in Base 64

Best Known (14, 53, s)-Nets in Base 64

(14, 53, 177)-Net over F₆₄ — Constructive and digital

Digital (14, 53, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 53, 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]

(14, 53, 257)-Net in Base 64 — Constructive

(14, 53, 257)-net in base 64, using

3 times m-reduction [i] based on (14, 56, 257)-net in base 64, using
- base change [i] based on digital (0, 42, 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]

(14, 53, 257)-Net over F₆₄ — Digital

Digital (14, 53, 257)-net over F₆₄, using

t-expansion [i] based on digital (12, 53, 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]

(14, 53, 11034)-Net in Base 64 — Upper bound on s

There is no (14, 53, 11035)-net in base 64, because

1 times m-reduction [i] would yield (14, 52, 11035)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 8356 059204 960842 939296 129534 019145 680609 577228 413064 642004 051436 183669 408167 717992 591213 162808 > 64⁵² [i]