MinT - Best Known (213−103, 213, s)-Nets in Base 3

Best Known (213−103, 213, s)-Nets in Base 3

(213−103, 213, 75)-Net over F₃ — Constructive and digital

Digital (110, 213, 75)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (27, 78, 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 (32, 135, 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]

(213−103, 213, 116)-Net over F₃ — Digital

Digital (110, 213, 116)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(213−103, 213, 905)-Net in Base 3 — Upper bound on s

There is no (110, 213, 906)-net in base 3, because

1 times m-reduction [i] would yield (110, 212, 906)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 141825 822645 854745 441464 797771 914145 254048 345399 590434 285419 429727 570705 935977 303547 809445 272820 661713 > 3²¹² [i]