MinT - Best Known (234−101, 234, s)-Nets in Base 4

Best Known (234−101, 234, s)-Nets in Base 4

(234−101, 234, 131)-Net over F₄ — Constructive and digital

Digital (133, 234, 131)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 60, 27)-net over F₄, using
  - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
      - an explicitly constructive algebraic function field [i]
2. digital (73, 174, 104)-net over F₄, using
  - net from sequence [i] based on digital (73, 103)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
      - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(234−101, 234, 263)-Net over F₄ — Digital

Digital (133, 234, 263)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(234−101, 234, 4109)-Net in Base 4 — Upper bound on s

There is no (133, 234, 4110)-net in base 4, because

1 times m-reduction [i] would yield (133, 233, 4110)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 190 846158 723557 743041 259631 719238 075005 741496 431586 215770 614725 889122 819824 386472 071575 847364 782427 851450 943533 585057 564222 880449 015074 157056 > 4²³³ [i]