MinT - Best Known (63, 63+59, s)-Nets in Base 3

Best Known (63, 63+59, s)-Nets in Base 3

(63, 63+59, 56)-Net over F₃ — Constructive and digital

Digital (63, 122, 56)-net over F₃, using

7 times m-reduction [i] based on digital (63, 129, 56)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 48, 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, 81, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(63, 63+59, 73)-Net over F₃ — Digital

Digital (63, 122, 73)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(63, 63+59, 543)-Net in Base 3 — Upper bound on s

There is no (63, 122, 544)-net in base 3, because

1 times m-reduction [i] would yield (63, 121, 544)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5573 226795 769487 963497 008531 406266 936012 314888 712043 425345 > 3¹²¹ [i]