MinT - Best Known (152, 152+77, s)-Nets in Base 4

Best Known (152, 152+77, s)-Nets in Base 4

(152, 152+77, 164)-Net over F₄ — Constructive and digital

Digital (152, 229, 164)-net over F₄, using

t-expansion [i] based on digital (151, 229, 164)-net over F₄, using
- 1 times m-reduction [i] based on digital (151, 230, 164)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (21, 60, 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 (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]

(152, 152+77, 240)-Net in Base 4 — Constructive

(152, 229, 240)-net in base 4, using

5 times m-reduction [i] based on (152, 234, 240)-net in base 4, using
- trace code for nets [i] based on (35, 117, 120)-net in base 16, using
  - 3 times m-reduction [i] based on (35, 120, 120)-net in base 16, using
    - base change [i] based on digital (11, 96, 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]

(152, 152+77, 600)-Net over F₄ — Digital

Digital (152, 229, 600)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(152, 152+77, 20482)-Net in Base 4 — Upper bound on s

There is no (152, 229, 20483)-net in base 4, because

1 times m-reduction [i] would yield (152, 228, 20483)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 186260 187538 121515 605913 821069 846710 451236 618913 970420 009466 232424 812513 422798 279425 516275 598193 147235 271536 814924 335999 669828 555778 708480 > 4²²⁸ [i]