MinT - Best Known (194−91, 194, s)-Nets in Base 3

Best Known (194−91, 194, s)-Nets in Base 3

(194−91, 194, 74)-Net over F₃ — Constructive and digital

Digital (103, 194, 74)-net over F₃, using

7 times m-reduction [i] based on digital (103, 201, 74)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (27, 76, 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, 125, 37)-net over F₃, using
    - net from sequence [i] based on digital (27, 36)-sequence over F₃ (see above)

(194−91, 194, 117)-Net over F₃ — Digital

Digital (103, 194, 117)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(194−91, 194, 936)-Net in Base 3 — Upper bound on s

There is no (103, 194, 937)-net in base 3, because

1 times m-reduction [i] would yield (103, 193, 937)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 122 244318 076607 674595 938816 535272 874418 311574 037853 303626 730842 939891 009106 902812 399127 544483 > 3¹⁹³ [i]