MinT - Best Known (84, 84+135, s)-Nets in Base 4

Best Known (84, 84+135, s)-Nets in Base 4

(84, 84+135, 104)-Net over F₄ — Constructive and digital

Digital (84, 219, 104)-net over F₄, using

t-expansion [i] based on digital (73, 219, 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]

(84, 84+135, 129)-Net over F₄ — Digital

Digital (84, 219, 129)-net over F₄, using

t-expansion [i] based on digital (81, 219, 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]

(84, 84+135, 728)-Net in Base 4 — Upper bound on s

There is no (84, 219, 729)-net in base 4, because

1 times m-reduction [i] would yield (84, 218, 729)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 191980 093467 935223 844285 224271 751128 544563 564116 730150 100649 426523 990776 210066 493149 852065 136173 298904 037568 686897 567357 319035 789560 > 4²¹⁸ [i]