MinT - Best Known (233−77, 233, s)-Nets in Base 4

Best Known (233−77, 233, s)-Nets in Base 4

(233−77, 233, 225)-Net over F₄ — Constructive and digital

Digital (156, 233, 225)-net over F₄, using

base reduction for projective spaces (embedding PG(116,16) in PG(232,4)) for nets [i] based on digital (40, 117, 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]

(233−77, 233, 240)-Net in Base 4 — Constructive

(156, 233, 240)-net in base 4, using

t-expansion [i] based on (155, 233, 240)-net in base 4, using
- 7 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]

(233−77, 233, 649)-Net over F₄ — Digital

Digital (156, 233, 649)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(233−77, 233, 23705)-Net in Base 4 — Upper bound on s

There is no (156, 233, 23706)-net in base 4, because

1 times m-reduction [i] would yield (156, 232, 23706)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 47 684113 375340 774170 429349 476297 573879 976675 171277 298704 437695 975979 235049 012426 564783 393419 824061 530114 729839 964419 429616 535047 130801 070000 > 4²³² [i]