MinT - Best Known (138−72, 138, s)-Nets in Base 3

Best Known (138−72, 138, s)-Nets in Base 3

(138−72, 138, 56)-Net over F₃ — Constructive and digital

Digital (66, 138, 56)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 51, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]
2. digital (15, 87, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(138−72, 138, 65)-Net over F₃ — Digital

Digital (66, 138, 65)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(138−72, 138, 447)-Net in Base 3 — Upper bound on s

There is no (66, 138, 448)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 742821 367550 517070 636381 662675 419811 766083 056989 307236 774017 151489 > 3¹³⁸ [i]