MinT - Best Known (222−75, 222, s)-Nets in Base 4

Best Known (222−75, 222, s)-Nets in Base 4

(222−75, 222, 164)-Net over F₄ — Constructive and digital

Digital (147, 222, 164)-net over F₄, using

1 times m-reduction [i] based on digital (147, 223, 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 (88, 164, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 82, 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]

(222−75, 222, 240)-Net in Base 4 — Constructive

(147, 222, 240)-net in base 4, using

4 times m-reduction [i] based on (147, 226, 240)-net in base 4, using
- trace code for nets [i] based on (34, 113, 120)-net in base 16, using
  - 2 times m-reduction [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]

(222−75, 222, 574)-Net over F₄ — Digital

Digital (147, 222, 574)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(222−75, 222, 19239)-Net in Base 4 — Upper bound on s

There is no (147, 222, 19240)-net in base 4, because

1 times m-reduction [i] would yield (147, 221, 19240)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11 366733 809091 694523 067329 538771 936215 502039 392739 759891 931928 695149 480296 059574 891831 856090 940404 981400 591165 226660 574683 259586 882003 > 4²²¹ [i]