MinT - Best Known (240−87, 240, s)-Nets in Base 4

Best Known (240−87, 240, s)-Nets in Base 4

(240−87, 240, 160)-Net over F₄ — Constructive and digital

Digital (153, 240, 160)-net over F₄, using

7 times m-reduction [i] based on digital (153, 247, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (33, 80, 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, 167, 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]

(240−87, 240, 208)-Net in Base 4 — Constructive

(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]

(240−87, 240, 480)-Net over F₄ — Digital

Digital (153, 240, 480)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(240−87, 240, 12458)-Net in Base 4 — Upper bound on s

There is no (153, 240, 12459)-net in base 4, because

1 times m-reduction [i] would yield (153, 239, 12459)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 782839 878066 581519 748836 238279 841865 300799 282941 710842 253296 977693 752007 377588 753683 112942 939544 663866 936338 162870 501508 623196 067406 922849 535548 > 4²³⁹ [i]