MinT - Best Known (223−79, 223, s)-Nets in Base 4

Best Known (223−79, 223, s)-Nets in Base 4

(223−79, 223, 160)-Net over F₄ — Constructive and digital

Digital (144, 223, 160)-net over F₄, using

4¹ times duplication [i] based on digital (143, 222, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 52, 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 (91, 170, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 85, 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]

(223−79, 223, 208)-Net in Base 4 — Constructive

(144, 223, 208)-net in base 4, using

1 times m-reduction [i] based on (144, 224, 208)-net in base 4, using
- trace code for nets [i] based on (32, 112, 104)-net in base 16, using
  - 3 times m-reduction [i] based on (32, 115, 104)-net in base 16, using
    - base change [i] based on digital (9, 92, 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]

(223−79, 223, 488)-Net over F₄ — Digital

Digital (144, 223, 488)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(223−79, 223, 13689)-Net in Base 4 — Upper bound on s

There is no (144, 223, 13690)-net in base 4, because

1 times m-reduction [i] would yield (144, 222, 13690)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 45 434710 442186 906036 666956 696519 370655 185467 438297 112343 743308 977848 697468 555789 138320 842705 862747 867573 670144 952918 413346 868806 550740 > 4²²² [i]