MinT - Best Known (236−87, 236, s)-Nets in Base 4

Best Known (236−87, 236, s)-Nets in Base 4

(236−87, 236, 160)-Net over F₄ — Constructive and digital

Digital (149, 236, 160)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (33, 76, 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, 160, 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]

(236−87, 236, 196)-Net in Base 4 — Constructive

(149, 236, 196)-net in base 4, using

trace code for nets [i] based on (31, 118, 98)-net in base 16, using
- 2 times m-reduction [i] based on (31, 120, 98)-net in base 16, using
  - base change [i] based on digital (7, 96, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(236−87, 236, 446)-Net over F₄ — Digital

Digital (149, 236, 446)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(236−87, 236, 10946)-Net in Base 4 — Upper bound on s

There is no (149, 236, 10947)-net in base 4, because

1 times m-reduction [i] would yield (149, 235, 10947)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3053 950962 013979 536275 196938 401585 094666 615584 592518 414369 173422 569526 796605 652622 287875 583963 266278 292082 996177 384151 376042 854373 024918 813280 > 4²³⁵ [i]