MinT - Best Known (240−89, 240, s)-Nets in Base 4

Best Known (240−89, 240, s)-Nets in Base 4

(240−89, 240, 160)-Net over F₄ — Constructive and digital

Digital (151, 240, 160)-net over F₄, using

1 times m-reduction [i] based on digital (151, 241, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (33, 78, 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, 163, 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]

(240−89, 240, 196)-Net in Base 4 — Constructive

(151, 240, 196)-net in base 4, using

trace code for nets [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]

(240−89, 240, 443)-Net over F₄ — Digital

Digital (151, 240, 443)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(240−89, 240, 10681)-Net in Base 4 — Upper bound on s

There is no (151, 240, 10682)-net in base 4, because

1 times m-reduction [i] would yield (151, 239, 10682)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 783348 341088 894848 613296 032724 148245 339645 544645 809009 352857 369031 032239 814085 436919 750897 364076 292761 247487 273238 886821 485293 990571 145011 021700 > 4²³⁹ [i]