MinT - Best Known (129, 129+115, s)-Nets in Base 3

Best Known (129, 129+115, s)-Nets in Base 3

(129, 129+115, 85)-Net over F₃ — Constructive and digital

Digital (129, 244, 85)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (27, 84, 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 (45, 160, 48)-net over F₃, using
  - net from sequence [i] based on digital (45, 47)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
      - an explicitly constructive algebraic function field [i]

(129, 129+115, 141)-Net over F₃ — Digital

Digital (129, 244, 141)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(129, 129+115, 1138)-Net in Base 3 — Upper bound on s

There is no (129, 244, 1139)-net in base 3, because

1 times m-reduction [i] would yield (129, 243, 1139)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 88 877722 131537 720353 616247 659141 343200 461612 538117 520604 427261 541922 450954 993113 832230 626012 950149 142176 636195 938231 > 3²⁴³ [i]