MinT - Best Known (212−89, 212, s)-Nets in Base 3

Best Known (212−89, 212, s)-Nets in Base 3

(212−89, 212, 148)-Net over F₃ — Constructive and digital

Digital (123, 212, 148)-net over F₃, using

trace code for nets [i] based on digital (17, 106, 74)-net over F₉, using
- net from sequence [i] based on digital (17, 73)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
    - a function field by GarcÃa and Quoos [i]

(212−89, 212, 172)-Net over F₃ — Digital

Digital (123, 212, 172)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(212−89, 212, 1631)-Net in Base 3 — Upper bound on s

There is no (123, 212, 1632)-net in base 3, because

1 times m-reduction [i] would yield (123, 211, 1632)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 47618 279602 469972 269864 385263 257995 543280 238546 867892 794299 746343 838197 850625 335223 688837 294869 100801 > 3²¹¹ [i]