MinT - Best Known (199−97, 199, s)-Nets in Base 3

Best Known (199−97, 199, s)-Nets in Base 3

(199−97, 199, 74)-Net over F₃ — Constructive and digital

Digital (102, 199, 74)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (27, 75, 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 (27, 124, 37)-net over F₃, using
  - net from sequence [i] based on digital (27, 36)-sequence over F₃ (see above)

(199−97, 199, 107)-Net over F₃ — Digital

Digital (102, 199, 107)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(199−97, 199, 824)-Net in Base 3 — Upper bound on s

There is no (102, 199, 825)-net in base 3, because

1 times m-reduction [i] would yield (102, 198, 825)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 30704 786741 038334 886514 920917 092554 389897 966596 414952 293167 462510 907202 654495 175814 511186 803137 > 3¹⁹⁸ [i]