MinT - Best Known (86, 227, s)-Nets in Base 4

Best Known (86, 227, s)-Nets in Base 4

(86, 227, 104)-Net over F₄ — Constructive and digital

Digital (86, 227, 104)-net over F₄, using

t-expansion [i] based on digital (73, 227, 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]

(86, 227, 129)-Net over F₄ — Digital

Digital (86, 227, 129)-net over F₄, using

t-expansion [i] based on digital (81, 227, 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]

(86, 227, 731)-Net in Base 4 — Upper bound on s

There is no (86, 227, 732)-net in base 4, because

1 times m-reduction [i] would yield (86, 226, 732)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12231 406944 333540 774954 252323 804535 730587 321117 033179 026691 945787 171000 516313 400458 227501 691189 176311 532945 901387 996319 346580 773710 867614 > 4²²⁶ [i]