MinT - Best Known (75−63, 75, s)-Nets in Base 64

Best Known (75−63, 75, s)-Nets in Base 64

(75−63, 75, 177)-Net over F₆₄ — Constructive and digital

Digital (12, 75, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 75, 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]

(75−63, 75, 257)-Net over F₆₄ — Digital

Digital (12, 75, 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]

(75−63, 75, 4023)-Net in Base 64 — Upper bound on s

There is no (12, 75, 4024)-net in base 64, because

1 times m-reduction [i] would yield (12, 74, 4024)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 45 774712 472856 943768 949475 010233 455435 803241 655962 011241 491038 325136 987274 126935 859706 189470 180012 905415 019764 083525 234908 969281 818812 > 64⁷⁴ [i]