MinT - Best Known (158, 158+77, s)-Nets in Base 3

Best Known (158, 158+77, s)-Nets in Base 3

(158, 158+77, 167)-Net over F₃ — Constructive and digital

Digital (158, 235, 167)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (9, 47, 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 (111, 188, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 94, 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]

(158, 158+77, 386)-Net over F₃ — Digital

Digital (158, 235, 386)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(158, 158+77, 6476)-Net in Base 3 — Upper bound on s

There is no (158, 235, 6477)-net in base 3, because

1 times m-reduction [i] would yield (158, 234, 6477)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 4445 013236 070346 767205 911513 140119 466181 309829 667830 767407 253351 440362 820958 016271 898234 802066 761940 244117 032849 > 3²³⁴ [i]