MinT - Best Known (198−67, 198, s)-Nets in Base 4

Best Known (198−67, 198, s)-Nets in Base 4

(198−67, 198, 163)-Net over F₄ — Constructive and digital

Digital (131, 198, 163)-net over F₄, using

t-expansion [i] based on digital (130, 198, 163)-net over F₄, using
- 1 times m-reduction [i] based on digital (130, 199, 163)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (15, 49, 33)-net over F₄, using
      - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
    2. digital (81, 150, 130)-net over F₄, using
      - trace code for nets [i] based on digital (6, 75, 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]

(198−67, 198, 240)-Net in Base 4 — Constructive

(131, 198, 240)-net in base 4, using

2 times m-reduction [i] based on (131, 200, 240)-net in base 4, using
- trace code for nets [i] based on (31, 100, 120)-net in base 16, using
  - base change [i] based on digital (11, 80, 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]

(198−67, 198, 516)-Net over F₄ — Digital

Digital (131, 198, 516)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(198−67, 198, 17205)-Net in Base 4 — Upper bound on s

There is no (131, 198, 17206)-net in base 4, because

1 times m-reduction [i] would yield (131, 197, 17206)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 40384 203532 578677 162361 016381 431829 795502 705610 570151 070295 539846 982482 055493 911777 791874 246828 195224 392871 923707 563629 > 4¹⁹⁷ [i]