MinT - Best Known (227−81, 227, s)-Nets in Base 4

Best Known (227−81, 227, s)-Nets in Base 4

(227−81, 227, 160)-Net over F₄ — Constructive and digital

Digital (146, 227, 160)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (33, 73, 56)-net over F₄, using
  - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
      - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
2. digital (73, 154, 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]

(227−81, 227, 208)-Net in Base 4 — Constructive

(146, 227, 208)-net in base 4, using

1 times m-reduction [i] based on (146, 228, 208)-net in base 4, using
- trace code for nets [i] based on (32, 114, 104)-net in base 16, using
  - 1 times m-reduction [i] based on (32, 115, 104)-net in base 16, using
    - base change [i] based on digital (9, 92, 104)-net over F₃₂, using
      - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(227−81, 227, 484)-Net over F₄ — Digital

Digital (146, 227, 484)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(227−81, 227, 13220)-Net in Base 4 — Upper bound on s

There is no (146, 227, 13221)-net in base 4, because

1 times m-reduction [i] would yield (146, 226, 13221)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11663 382151 361375 854993 345191 659459 999999 224768 468309 245896 511368 336442 730977 547196 949822 047007 446567 430334 620128 998229 401962 105528 943408 > 4²²⁶ [i]