MinT - Best Known (81, 81+162, s)-Nets in Base 4

Best Known (81, 81+162, s)-Nets in Base 4

(81, 81+162, 104)-Net over F₄ — Constructive and digital

Digital (81, 243, 104)-net over F₄, using

t-expansion [i] based on digital (73, 243, 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+162, 129)-Net over F₄ — Digital

Digital (81, 243, 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+162, 596)-Net in Base 4 — Upper bound on s

There is no (81, 243, 597)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 223 227073 878487 160640 275043 217902 860639 601777 757628 906975 856522 092278 974739 586980 140055 648838 554784 170210 509078 705165 376220 593367 921897 646611 752144 > 4²⁴³ [i]