MinT - Best Known (85, 85+166, s)-Nets in Base 4

Best Known (85, 85+166, s)-Nets in Base 4

(85, 85+166, 104)-Net over F₄ — Constructive and digital

Digital (85, 251, 104)-net over F₄, using

t-expansion [i] based on digital (73, 251, 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+166, 129)-Net over F₄ — Digital

Digital (85, 251, 129)-net over F₄, using

t-expansion [i] based on digital (81, 251, 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+166, 633)-Net in Base 4 — Upper bound on s

There is no (85, 251, 634)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 14 441655 486039 072652 533104 372530 373062 259438 849783 068051 644403 244352 633546 613249 331117 153459 489697 954397 096117 471977 349228 010455 412582 379220 214052 598036 > 4²⁵¹ [i]