MinT - Best Known (158, 158+81, s)-Nets in Base 4

Best Known (158, 158+81, s)-Nets in Base 4

(158, 158+81, 170)-Net over F₄ — Constructive and digital

Digital (158, 239, 170)-net over F₄, using

4¹ times duplication [i] based on digital (157, 238, 170)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (44, 84, 66)-net over F₄, using
    - trace code for nets [i] based on digital (2, 42, 33)-net over F₁₆, using
      - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
        a function field by SÃ©mirat [i]
  2. digital (73, 154, 104)-net over F₄, using
    - net from sequence [i] based on digital (73, 103)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
        F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(158, 158+81, 240)-Net in Base 4 — Constructive

(158, 239, 240)-net in base 4, using

5 times m-reduction [i] based on (158, 244, 240)-net in base 4, using
- trace code for nets [i] based on (36, 122, 120)-net in base 16, using
  - 3 times m-reduction [i] based on (36, 125, 120)-net in base 16, using
    - base change [i] based on digital (11, 100, 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]

(158, 158+81, 606)-Net over F₄ — Digital

Digital (158, 239, 606)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(158, 158+81, 20054)-Net in Base 4 — Upper bound on s

There is no (158, 239, 20055)-net in base 4, because

1 times m-reduction [i] would yield (158, 238, 20055)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 195214 108065 772632 902432 392212 206039 425920 263399 115820 518563 313353 755508 060092 112678 778266 755698 256084 462907 908838 028273 137599 339170 673933 563340 > 4²³⁸ [i]