MinT - Best Known (157, 157+78, s)-Nets in Base 4

Best Known (157, 157+78, s)-Nets in Base 4

(157, 157+78, 225)-Net over F₄ — Constructive and digital

Digital (157, 235, 225)-net over F₄, using

base reduction for projective spaces (embedding PG(117,16) in PG(234,4)) for nets [i] based on digital (40, 118, 225)-net over F₁₆, using
- net from sequence [i] based on digital (40, 224)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 40 and N(F) ≥ 225, using
    - a function field by GarcÃa and Quoos [i]

(157, 157+78, 240)-Net in Base 4 — Constructive

(157, 235, 240)-net in base 4, using

t-expansion [i] based on (155, 235, 240)-net in base 4, using
- 5 times m-reduction [i] based on (155, 240, 240)-net in base 4, using
  - trace code for nets [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]

(157, 157+78, 644)-Net over F₄ — Digital

Digital (157, 235, 644)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(157, 157+78, 21750)-Net in Base 4 — Upper bound on s

There is no (157, 235, 21751)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3052 950898 441961 253802 416038 315005 157527 372539 372896 613621 055383 648804 587837 183540 506025 883388 052550 455445 884975 406615 448746 903410 031350 783404 > 4²³⁵ [i]