MinT - Best Known (218−77, 218, s)-Nets in Base 4

Best Known (218−77, 218, s)-Nets in Base 4

(218−77, 218, 160)-Net over F₄ — Constructive and digital

Digital (141, 218, 160)-net over F₄, using

4¹ times duplication [i] based on digital (140, 217, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 51, 30)-net over F₄, using
    - net from sequence [i] based on digital (13, 29)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
        F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
  2. digital (89, 166, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 83, 65)-net over F₁₆, using
      - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(218−77, 218, 208)-Net in Base 4 — Constructive

(141, 218, 208)-net in base 4, using

2 times m-reduction [i] based on (141, 220, 208)-net in base 4, using
- trace code for nets [i] based on (31, 110, 104)-net in base 16, using
  - base change [i] based on digital (9, 88, 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]

(218−77, 218, 483)-Net over F₄ — Digital

Digital (141, 218, 483)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(218−77, 218, 13701)-Net in Base 4 — Upper bound on s

There is no (141, 218, 13702)-net in base 4, because

1 times m-reduction [i] would yield (141, 217, 13702)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 44395 698691 612090 680149 321504 891624 362887 506144 218182 709325 440792 336820 616613 876276 275440 826445 636870 877643 820781 668738 266516 916384 > 4²¹⁷ [i]