MinT - Best Known (217−127, 217, s)-Nets in Base 4

Best Known (217−127, 217, s)-Nets in Base 4

(217−127, 217, 104)-Net over F₄ — Constructive and digital

Digital (90, 217, 104)-net over F₄, using

t-expansion [i] based on digital (73, 217, 104)-net over F₄, using
- net from sequence [i] based on digital (73, 103)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
    - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(217−127, 217, 129)-Net over F₄ — Digital

Digital (90, 217, 129)-net over F₄, using

t-expansion [i] based on digital (81, 217, 129)-net over F₄, using
- net from sequence [i] based on digital (81, 128)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 81 and N(F) ≥ 129, using
    - Drinfeld modules of rank 1 [i]

(217−127, 217, 888)-Net in Base 4 — Upper bound on s

There is no (90, 217, 889)-net in base 4, because

1 times m-reduction [i] would yield (90, 216, 889)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11654 262368 235253 700553 901055 353493 174747 522737 126721 378077 626207 600214 310687 806662 302270 442488 401344 674058 615489 438907 818752 549056 > 4²¹⁶ [i]