MinT - Best Known (197−75, 197, s)-Nets in Base 4

Best Known (197−75, 197, s)-Nets in Base 4

(197−75, 197, 132)-Net over F₄ — Constructive and digital

Digital (122, 197, 132)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (12, 49, 28)-net over F₄, using
  - net from sequence [i] based on digital (12, 27)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 12 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]
2. digital (73, 148, 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]

(197−75, 197, 341)-Net over F₄ — Digital

Digital (122, 197, 341)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(197−75, 197, 7522)-Net in Base 4 — Upper bound on s

There is no (122, 197, 7523)-net in base 4, because

1 times m-reduction [i] would yield (122, 196, 7523)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 10133 988847 064736 598657 893252 140319 305819 503401 931312 801086 797154 314960 595967 071531 152358 503279 830852 067665 283516 096032 > 4¹⁹⁶ [i]