MinT - Best Known (126, 126+45, s)-Nets in Base 4

Best Known (126, 126+45, s)-Nets in Base 4

(126, 126+45, 531)-Net over F₄ — Constructive and digital

Digital (126, 171, 531)-net over F₄, using

t-expansion [i] based on digital (125, 171, 531)-net over F₄, using
- 6 times m-reduction [i] based on digital (125, 177, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 59, 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]

(126, 126+45, 576)-Net in Base 4 — Constructive

(126, 171, 576)-net in base 4, using

4³ times duplication [i] based on (123, 168, 576)-net in base 4, using
- trace code for nets [i] based on (11, 56, 192)-net in base 64, using
  - base change [i] based on digital (3, 48, 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]

(126, 126+45, 1280)-Net over F₄ — Digital

Digital (126, 171, 1280)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(126, 126+45, 135505)-Net in Base 4 — Upper bound on s

There is no (126, 171, 135506)-net in base 4, because

1 times m-reduction [i] would yield (126, 170, 135506)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 239988 808074 335976 091875 227753 912369 880731 999018 990869 088096 939852 942599 822006 965428 278458 188868 783584 > 4¹⁷⁰ [i]