MinT - Best Known (240−86, 240, s)-Nets in Base 4

Best Known (240−86, 240, s)-Nets in Base 4

(240−86, 240, 160)-Net over F₄ — Constructive and digital

Digital (154, 240, 160)-net over F₄, using

10 times m-reduction [i] based on digital (154, 250, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (33, 81, 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, 169, 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]

(240−86, 240, 208)-Net in Base 4 — Constructive

(154, 240, 208)-net in base 4, using

t-expansion [i] based on (153, 240, 208)-net in base 4, using
- trace code for nets [i] based on (33, 120, 104)-net in base 16, using
  - base change [i] based on digital (9, 96, 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]

(240−86, 240, 500)-Net over F₄ — Digital

Digital (154, 240, 500)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(240−86, 240, 12867)-Net in Base 4 — Upper bound on s

There is no (154, 240, 12868)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 127491 396607 128975 666365 186455 951612 269875 905655 708655 497748 842750 228321 445654 777239 849974 100746 083588 011648 347991 005961 824122 390130 712648 884120 > 4²⁴⁰ [i]