MinT - Best Known (245−81, 245, s)-Nets in Base 3

Best Known (245−81, 245, s)-Nets in Base 3

(245−81, 245, 167)-Net over F₃ — Constructive and digital

Digital (164, 245, 167)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (9, 49, 19)-net over F₃, using
  - net from sequence [i] based on digital (9, 18)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 9 and N(F) ≥ 19, using
      - an explicitly constructive algebraic function field [i]
2. digital (115, 196, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 98, 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]

(245−81, 245, 389)-Net over F₃ — Digital

Digital (164, 245, 389)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(245−81, 245, 6375)-Net in Base 3 — Upper bound on s

There is no (164, 245, 6376)-net in base 3, because

1 times m-reduction [i] would yield (164, 244, 6376)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 262 120545 406158 583577 838166 632361 822495 695246 143001 698552 907568 834577 374337 367807 637224 452652 087654 609678 176823 768321 > 3²⁴⁴ [i]