MinT - Best Known (218−103, 218, s)-Nets in Base 3

Best Known (218−103, 218, s)-Nets in Base 3

(218−103, 218, 76)-Net over F₃ — Constructive and digital

Digital (115, 218, 76)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (32, 83, 38)-net over F₃, using
  - net from sequence [i] based on digital (32, 37)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 32 and N(F) ≥ 38, using
      - an explicitly constructive algebraic function field [i]
2. digital (32, 135, 38)-net over F₃, using
  - net from sequence [i] based on digital (32, 37)-sequence over F₃ (see above)

(218−103, 218, 126)-Net over F₃ — Digital

Digital (115, 218, 126)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(218−103, 218, 1014)-Net in Base 3 — Upper bound on s

There is no (115, 218, 1015)-net in base 3, because

1 times m-reduction [i] would yield (115, 217, 1015)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 35 225768 238848 098884 459803 193003 911453 376303 852976 854700 412706 150361 745838 236461 768812 225729 590599 562619 > 3²¹⁷ [i]