MinT - Best Known (164, 164+87, s)-Nets in Base 4

Best Known (164, 164+87, s)-Nets in Base 4

(164, 164+87, 200)-Net over F₄ — Constructive and digital

Digital (164, 251, 200)-net over F₄, using

t-expansion [i] based on digital (161, 251, 200)-net over F₄, using
- net from sequence [i] based on digital (161, 199)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 161 and N(F) ≥ 200, using
    - F₇ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(164, 164+87, 240)-Net in Base 4 — Constructive

(164, 251, 240)-net in base 4, using

3 times m-reduction [i] based on (164, 254, 240)-net in base 4, using
- trace code for nets [i] based on (37, 127, 120)-net in base 16, using
  - 3 times m-reduction [i] based on (37, 130, 120)-net in base 16, using
    - base change [i] based on digital (11, 104, 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]

(164, 164+87, 585)-Net over F₄ — Digital

Digital (164, 251, 585)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(164, 164+87, 17776)-Net in Base 4 — Upper bound on s

There is no (164, 251, 17777)-net in base 4, because

1 times m-reduction [i] would yield (164, 250, 17777)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 281210 931307 332813 315375 443989 798800 364755 443103 781537 742021 137888 920717 878399 336329 666200 932046 716511 938761 868307 682661 891521 503168 178960 156980 098704 > 4²⁵⁰ [i]