MinT - Best Known (197−51, 197, s)-Nets in Base 4

Best Known (197−51, 197, s)-Nets in Base 4

(197−51, 197, 531)-Net over F₄ — Constructive and digital

Digital (146, 197, 531)-net over F₄, using

t-expansion [i] based on digital (145, 197, 531)-net over F₄, using
- 10 times m-reduction [i] based on digital (145, 207, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 69, 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]

(197−51, 197, 576)-Net in Base 4 — Constructive

(146, 197, 576)-net in base 4, using

t-expansion [i] based on (145, 197, 576)-net in base 4, using
- 1 times m-reduction [i] based on (145, 198, 576)-net in base 4, using
  - trace code for nets [i] based on (13, 66, 192)-net in base 64, using
    - 4 times m-reduction [i] based on (13, 70, 192)-net in base 64, using
      - base change [i] based on digital (3, 60, 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]

(197−51, 197, 1555)-Net over F₄ — Digital

Digital (146, 197, 1555)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(197−51, 197, 178075)-Net in Base 4 — Upper bound on s

There is no (146, 197, 178076)-net in base 4, because

1 times m-reduction [i] would yield (146, 196, 178076)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 10086 955176 438076 764229 517744 519024 444496 244459 100761 016297 325259 410778 930618 317385 243746 552149 663392 819316 606100 447913 > 4¹⁹⁶ [i]