MinT - Best Known (81, 81+172, s)-Nets in Base 4

Best Known (81, 81+172, s)-Nets in Base 4

(81, 81+172, 104)-Net over F₄ — Constructive and digital

Digital (81, 253, 104)-net over F₄, using

t-expansion [i] based on digital (73, 253, 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]

(81, 81+172, 129)-Net over F₄ — Digital

Digital (81, 253, 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]

(81, 81+172, 577)-Net in Base 4 — Upper bound on s

There is no (81, 253, 578)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 222 753063 156970 886001 585956 203683 334814 100738 425795 139695 027795 256477 729670 933205 348500 543998 596250 851397 409970 769402 989694 768334 565776 368174 197800 134400 > 4²⁵³ [i]