MinT - Best Known (224−75, 224, s)-Nets in Base 4

Best Known (224−75, 224, s)-Nets in Base 4

(224−75, 224, 164)-Net over F₄ — Constructive and digital

Digital (149, 224, 164)-net over F₄, using

t-expansion [i] based on digital (148, 224, 164)-net over F₄, using
- 1 times m-reduction [i] based on digital (148, 225, 164)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (21, 59, 34)-net over F₄, using
      - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
        T₅ from the second 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]

(224−75, 224, 240)-Net in Base 4 — Constructive

(149, 224, 240)-net in base 4, using

6 times m-reduction [i] based on (149, 230, 240)-net in base 4, using
- trace code for nets [i] based on (34, 115, 120)-net in base 16, using
  - base change [i] based on digital (11, 92, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]

(224−75, 224, 598)-Net over F₄ — Digital

Digital (149, 224, 598)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(224−75, 224, 20739)-Net in Base 4 — Upper bound on s

There is no (149, 224, 20740)-net in base 4, because

1 times m-reduction [i] would yield (149, 223, 20740)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 182 030712 645750 778339 822576 202528 624359 294229 752972 841165 295727 713289 306719 734875 607456 605695 164306 113206 151459 698961 457025 771153 122368 > 4²²³ [i]