MinT - Best Known (191−89, 191, s)-Nets in Base 3

Best Known (191−89, 191, s)-Nets in Base 3

(191−89, 191, 74)-Net over F₃ — Constructive and digital

Digital (102, 191, 74)-net over F₃, using

7 times m-reduction [i] based on digital (102, 198, 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, 123, 37)-net over F₃, using
    - net from sequence [i] based on digital (27, 36)-sequence over F₃ (see above)

(191−89, 191, 117)-Net over F₃ — Digital

Digital (102, 191, 117)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(191−89, 191, 948)-Net in Base 3 — Upper bound on s

There is no (102, 191, 949)-net in base 3, because

1 times m-reduction [i] would yield (102, 190, 949)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 4 590823 473447 209119 253386 134377 284931 330606 854803 863409 688810 284090 238371 820885 485472 362801 > 3¹⁹⁰ [i]