MinT - Best Known (219−91, 219, s)-Nets in Base 4

Best Known (219−91, 219, s)-Nets in Base 4

(219−91, 219, 131)-Net over F₄ — Constructive and digital

Digital (128, 219, 131)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 55, 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, 164, 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]

(219−91, 219, 280)-Net over F₄ — Digital

Digital (128, 219, 280)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−91, 219, 4812)-Net in Base 4 — Upper bound on s

There is no (128, 219, 4813)-net in base 4, because

1 times m-reduction [i] would yield (128, 218, 4813)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 178069 270073 696378 986264 045131 075493 782594 879355 930897 957274 208004 939872 894416 475774 350414 446173 796310 940276 298963 920780 692180 412040 > 4²¹⁸ [i]