MinT - Best Known (12, 83, s)-Nets in Base 64

Best Known (12, 83, s)-Nets in Base 64

(12, 83, 177)-Net over F₆₄ — Constructive and digital

Digital (12, 83, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 83, 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]

(12, 83, 257)-Net over F₆₄ — Digital

Digital (12, 83, 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]

(12, 83, 3745)-Net in Base 64 — Upper bound on s

There is no (12, 83, 3746)-net in base 64, because

1 times m-reduction [i] would yield (12, 82, 3746)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 12839 945493 530207 652158 419720 967191 371450 308891 526811 454117 924079 071233 355172 051021 537701 803031 382917 483378 123849 127638 016435 634964 896498 088506 246336 > 64⁸² [i]