MinT - Best Known (126, 126+103, s)-Nets in Base 3

Best Known (126, 126+103, s)-Nets in Base 3

(126, 126+103, 85)-Net over F₃ — Constructive and digital

Digital (126, 229, 85)-net over F₃, using

5 times m-reduction [i] based on digital (126, 234, 85)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (27, 81, 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, 153, 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]

(126, 126+103, 152)-Net over F₃ — Digital

Digital (126, 229, 152)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(126, 126+103, 1298)-Net in Base 3 — Upper bound on s

There is no (126, 229, 1299)-net in base 3, because

1 times m-reduction [i] would yield (126, 228, 1299)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 6 137835 202670 462709 306553 994536 058715 336117 421317 215125 759971 531518 242866 765945 174117 318358 884057 137455 209099 > 3²²⁸ [i]