MinT - Best Known (237−87, 237, s)-Nets in Base 4

Best Known (237−87, 237, s)-Nets in Base 4

(237−87, 237, 160)-Net over F₄ — Constructive and digital

Digital (150, 237, 160)-net over F₄, using

1 times m-reduction [i] based on digital (150, 238, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (33, 77, 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, 161, 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]

(237−87, 237, 196)-Net in Base 4 — Constructive

(150, 237, 196)-net in base 4, using

1 times m-reduction [i] based on (150, 238, 196)-net in base 4, using
- trace code for nets [i] based on (31, 119, 98)-net in base 16, using
  - 1 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]

(237−87, 237, 454)-Net over F₄ — Digital

Digital (150, 237, 454)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(237−87, 237, 11306)-Net in Base 4 — Upper bound on s

There is no (150, 237, 11307)-net in base 4, because

1 times m-reduction [i] would yield (150, 236, 11307)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12223 664944 199476 873497 333764 846537 761516 660922 852625 198154 825723 865305 513146 312531 558960 241073 957403 939725 635215 575128 844317 041038 430664 417900 > 4²³⁶ [i]