MinT - Best Known (131−33, 131, s)-Nets in Base 4

Best Known (131−33, 131, s)-Nets in Base 4

(131−33, 131, 531)-Net over F₄ — Constructive and digital

Digital (98, 131, 531)-net over F₄, using

t-expansion [i] based on digital (97, 131, 531)-net over F₄, using
- 4 times m-reduction [i] based on digital (97, 135, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 45, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(131−33, 131, 576)-Net in Base 4 — Constructive

(98, 131, 576)-net in base 4, using

1 times m-reduction [i] based on (98, 132, 576)-net in base 4, using
- trace code for nets [i] based on (10, 44, 192)-net in base 64, using
  - 5 times m-reduction [i] based on (10, 49, 192)-net in base 64, using
    - base change [i] based on digital (3, 42, 192)-net over F₁₂₈, using
      - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(131−33, 131, 1259)-Net over F₄ — Digital

Digital (98, 131, 1259)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(131−33, 131, 176653)-Net in Base 4 — Upper bound on s

There is no (98, 131, 176654)-net in base 4, because

1 times m-reduction [i] would yield (98, 130, 176654)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 852681 909971 848428 374872 633657 557119 804569 962446 755178 161584 205769 568874 278432 > 4¹³⁰ [i]