MinT - Best Known (84−71, 84, s)-Nets in Base 64

Best Known (84−71, 84, s)-Nets in Base 64

(84−71, 84, 177)-Net over F₆₄ — Constructive and digital

Digital (13, 84, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 84, 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]

(84−71, 84, 257)-Net over F₆₄ — Digital

Digital (13, 84, 257)-net over F₆₄, using

t-expansion [i] based on digital (12, 84, 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]

(84−71, 84, 4220)-Net in Base 64 — Upper bound on s

There is no (13, 84, 4221)-net in base 64, because

1 times m-reduction [i] would yield (13, 83, 4221)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 822737 196523 443607 597969 186381 740079 650387 464098 703471 634741 186288 247770 850446 328799 574134 293064 124102 213730 270912 155412 634563 962369 685774 045849 887748 > 64⁸³ [i]