MinT - Best Known (227−95, 227, s)-Nets in Base 4

Best Known (227−95, 227, s)-Nets in Base 4

(227−95, 227, 132)-Net over F₄ — Constructive and digital

Digital (132, 227, 132)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (12, 59, 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, 168, 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]

(227−95, 227, 283)-Net over F₄ — Digital

Digital (132, 227, 283)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(227−95, 227, 4769)-Net in Base 4 — Upper bound on s

There is no (132, 227, 4770)-net in base 4, because

1 times m-reduction [i] would yield (132, 226, 4770)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11659 714281 694893 997223 610568 749693 276293 023448 254195 435512 300303 860670 092394 447753 165800 147716 321891 730945 560549 216813 269682 917470 529856 > 4²²⁶ [i]