MinT - Best Known (13, 13+75, s)-Nets in Base 64

Best Known (13, 13+75, s)-Nets in Base 64

(13, 13+75, 177)-Net over F₆₄ — Constructive and digital

Digital (13, 88, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 88, 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]

(13, 13+75, 257)-Net over F₆₄ — Digital

Digital (13, 88, 257)-net over F₆₄, using

t-expansion [i] based on digital (12, 88, 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]

(13, 13+75, 4088)-Net in Base 64 — Upper bound on s

There is no (13, 88, 4089)-net in base 64, because

1 times m-reduction [i] would yield (13, 87, 4089)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 13 788298 218547 331595 225331 615455 717306 772110 500165 821517 741625 479734 441271 122857 592269 853415 868956 870568 551773 042382 307331 033388 293596 043960 267190 546849 959480 > 64⁸⁷ [i]