MinT - Best Known (85, 85+138, s)-Nets in Base 4

Best Known (85, 85+138, s)-Nets in Base 4

(85, 85+138, 104)-Net over F₄ — Constructive and digital

Digital (85, 223, 104)-net over F₄, using

t-expansion [i] based on digital (73, 223, 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]

(85, 85+138, 129)-Net over F₄ — Digital

Digital (85, 223, 129)-net over F₄, using

t-expansion [i] based on digital (81, 223, 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]

(85, 85+138, 724)-Net in Base 4 — Upper bound on s

There is no (85, 223, 725)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 182 229388 410338 394121 801274 806741 942918 564476 457750 813868 002352 493046 634818 997843 613197 874598 145420 097995 020628 597006 048293 384700 656060 > 4²²³ [i]