MinT - Best Known (189−90, 189, s)-Nets in Base 3

Best Known (189−90, 189, s)-Nets in Base 3

(189−90, 189, 74)-Net over F₃ — Constructive and digital

Digital (99, 189, 74)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (27, 72, 37)-net over F₃, using
  - net from sequence [i] based on digital (27, 36)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 26, N(F)Â =Â 36, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using an explicitly constructive algebraic function field [i]
2. digital (27, 117, 37)-net over F₃, using
  - net from sequence [i] based on digital (27, 36)-sequence over F₃ (see above)

(189−90, 189, 109)-Net over F₃ — Digital

Digital (99, 189, 109)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(189−90, 189, 845)-Net in Base 3 — Upper bound on s

There is no (99, 189, 846)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 521314 758470 591852 856519 173327 858501 191215 978913 605539 919217 029439 997671 423539 959063 131109 > 3¹⁸⁹ [i]