MinT - Best Known (86, 86+139, s)-Nets in Base 4

Best Known (86, 86+139, s)-Nets in Base 4

(86, 86+139, 104)-Net over F₄ — Constructive and digital

Digital (86, 225, 104)-net over F₄, using

t-expansion [i] based on digital (73, 225, 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, 86+139, 129)-Net over F₄ — Digital

Digital (86, 225, 129)-net over F₄, using

t-expansion [i] based on digital (81, 225, 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, 86+139, 740)-Net in Base 4 — Upper bound on s

There is no (86, 225, 741)-net in base 4, because

1 times m-reduction [i] would yield (86, 224, 741)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 741 050255 945832 915989 201070 900536 463728 205947 215209 706680 070716 966213 174705 916555 351667 832826 745536 066829 209592 850095 528953 710289 291732 > 4²²⁴ [i]