MinT - Best Known (172−87, 172, s)-Nets in Base 3

Best Known (172−87, 172, s)-Nets in Base 3

(172−87, 172, 65)-Net over F₃ — Constructive and digital

Digital (85, 172, 65)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 58, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]
2. digital (27, 114, 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]

(172−87, 172, 85)-Net over F₃ — Digital

Digital (85, 172, 85)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−87, 172, 625)-Net in Base 3 — Upper bound on s

There is no (85, 172, 626)-net in base 3, because

1 times m-reduction [i] would yield (85, 171, 626)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 4092 624623 449377 664750 572035 123802 390894 438772 968427 065349 743270 232797 483383 336577 > 3¹⁷¹ [i]