MinT - Best Known (203−65, 203, s)-Nets in Base 3

Best Known (203−65, 203, s)-Nets in Base 3

(203−65, 203, 164)-Net over F₃ — Constructive and digital

Digital (138, 203, 164)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (7, 39, 16)-net over F₃, using
  - net from sequence [i] based on digital (7, 15)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 7 and N(F) ≥ 16, using
      - an explicitly constructive algebraic function field [i]
2. digital (99, 164, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 82, 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]

(203−65, 203, 366)-Net over F₃ — Digital

Digital (138, 203, 366)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(203−65, 203, 6540)-Net in Base 3 — Upper bound on s

There is no (138, 203, 6541)-net in base 3, because

1 times m-reduction [i] would yield (138, 202, 6541)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 399021 366867 598142 601279 267280 669534 714236 916208 682503 541314 125498 116567 177232 047940 183797 435265 > 3²⁰² [i]