MinT - Best Known (251−111, 251, s)-Nets in Base 4

Best Known (251−111, 251, s)-Nets in Base 4

(251−111, 251, 132)-Net over F₄ — Constructive and digital

Digital (140, 251, 132)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (12, 67, 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, 184, 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]

(251−111, 251, 259)-Net over F₄ — Digital

Digital (140, 251, 259)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(251−111, 251, 3833)-Net in Base 4 — Upper bound on s

There is no (140, 251, 3834)-net in base 4, because

1 times m-reduction [i] would yield (140, 250, 3834)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 299050 169042 862566 724952 915182 322337 757732 108811 912744 052100 968822 804240 526550 071987 737130 999543 506919 261799 851383 702446 958606 291864 865377 731570 245340 > 4²⁵⁰ [i]