MinT - Best Known (137, 137+49, s)-Nets in Base 3

Best Known (137, 137+49, s)-Nets in Base 3

(137, 137+49, 288)-Net over F₃ — Constructive and digital

Digital (137, 186, 288)-net over F₃, using

3 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]

(137, 137+49, 654)-Net over F₃ — Digital

Digital (137, 186, 654)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 137+49, 23318)-Net in Base 3 — Upper bound on s

There is no (137, 186, 23319)-net in base 3, because

1 times m-reduction [i] would yield (137, 185, 23319)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 18515 976872 799157 582326 675504 999949 122152 173396 013928 308257 056797 691661 719424 905595 730769 > 3¹⁸⁵ [i]