MinT - Best Known (135−70, 135, s)-Nets in Base 3

Best Known (135−70, 135, s)-Nets in Base 3

(135−70, 135, 56)-Net over F₃ — Constructive and digital

Digital (65, 135, 56)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 50, 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 (15, 85, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(135−70, 135, 65)-Net over F₃ — Digital

Digital (65, 135, 65)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(135−70, 135, 447)-Net in Base 3 — Upper bound on s

There is no (65, 135, 448)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 25876 499247 424324 637383 179339 641245 434384 128595 902400 369803 169537 > 3¹³⁵ [i]