MinT - Best Known (162−77, 162, s)-Nets in Base 3

Best Known (162−77, 162, s)-Nets in Base 3

(162−77, 162, 68)-Net over F₃ — Constructive and digital

Digital (85, 162, 68)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (21, 59, 32)-net over F₃, using
  - net from sequence [i] based on digital (21, 31)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 21 and N(F) ≥ 32, using
      - an explicitly constructive algebraic function field [i]
2. digital (26, 103, 36)-net over F₃, using
  - net from sequence [i] based on digital (26, 35)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using
      - an explicitly constructive algebraic function field [i]

(162−77, 162, 96)-Net over F₃ — Digital

Digital (85, 162, 96)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(162−77, 162, 752)-Net in Base 3 — Upper bound on s

There is no (85, 162, 753)-net in base 3, because

1 times m-reduction [i] would yield (85, 161, 753)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 66951 731486 239926 626137 178560 545098 232222 303251 730130 827040 283243 400547 344057 > 3¹⁶¹ [i]