MinT - Best Known (138, 138+49, s)-Nets in Base 3

Best Known (138, 138+49, s)-Nets in Base 3

(138, 138+49, 288)-Net over F₃ — Constructive and digital

Digital (138, 187, 288)-net over F₃, using

t-expansion [i] based on digital (137, 187, 288)-net over F₃, using
- 2 times m-reduction [i] based on digital (137, 189, 288)-net over F₃, using
  - trace code for nets [i] based on digital (11, 63, 96)-net over F₂₇, using
    - net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
        an explicitly constructive algebraic function field [i]

(138, 138+49, 671)-Net over F₃ — Digital

Digital (138, 187, 671)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(138, 138+49, 24412)-Net in Base 3 — Upper bound on s

There is no (138, 187, 24413)-net in base 3, because

1 times m-reduction [i] would yield (138, 186, 24413)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 55583 547626 577294 711251 883751 298536 864094 607698 772138 369554 496291 332230 439082 412052 775201 > 3¹⁸⁶ [i]