MinT - Best Known (136, 136+107, s)-Nets in Base 4

Best Known (136, 136+107, s)-Nets in Base 4

(136, 136+107, 131)-Net over F₄ — Constructive and digital

Digital (136, 243, 131)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 63, 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, 180, 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]

(136, 136+107, 255)-Net over F₄ — Digital

Digital (136, 243, 255)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(136, 136+107, 3808)-Net in Base 4 — Upper bound on s

There is no (136, 243, 3809)-net in base 4, because

1 times m-reduction [i] would yield (136, 242, 3809)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 50 203496 812254 658127 648989 018940 844087 328626 322269 853792 027966 952508 466017 132779 617837 093728 034468 947048 604176 951795 156408 900072 959845 480756 134016 > 4²⁴² [i]