MinT - Best Known (216−89, 216, s)-Nets in Base 4

Best Known (216−89, 216, s)-Nets in Base 4

(216−89, 216, 131)-Net over F₄ — Constructive and digital

Digital (127, 216, 131)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 54, 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, 162, 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]

(216−89, 216, 284)-Net over F₄ — Digital

Digital (127, 216, 284)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(216−89, 216, 4995)-Net in Base 4 — Upper bound on s

There is no (127, 216, 4996)-net in base 4, because

1 times m-reduction [i] would yield (127, 215, 4996)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2789 844040 888671 361814 470612 696070 910812 783550 160421 202851 544494 792972 635564 590469 798544 417172 444134 795628 449967 692758 959113 727136 > 4²¹⁵ [i]