MinT - Best Known (55, 55+51, s)-Nets in Base 3

Best Known (55, 55+51, s)-Nets in Base 3

(55, 55+51, 56)-Net over F₃ — Constructive and digital

Digital (55, 106, 56)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 40, 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, 66, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(55, 55+51, 66)-Net over F₃ — Digital

Digital (55, 106, 66)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(55, 55+51, 489)-Net in Base 3 — Upper bound on s

There is no (55, 106, 490)-net in base 3, because

1 times m-reduction [i] would yield (55, 105, 490)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 128 733956 522350 245823 216509 400522 088156 504218 869861 > 3¹⁰⁵ [i]