MinT - Best Known (146, 146+50, s)-Nets in Base 4

Best Known (146, 146+50, s)-Nets in Base 4

(146, 146+50, 536)-Net over F₄ — Constructive and digital

Digital (146, 196, 536)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 25, 5)-net over F₄, using
  - net from sequence [i] based on digital (0, 4)-sequence over F₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 0 and N(F) ≥ 5, using
      - the rational function field F₄(x) [i]
    - Niederreiter sequence [i]
2. digital (121, 171, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 57, 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]

(146, 146+50, 648)-Net in Base 4 — Constructive

(146, 196, 648)-net in base 4, using

4¹ times duplication [i] based on (145, 195, 648)-net in base 4, using
- trace code for nets [i] based on (15, 65, 216)-net in base 64, using
  - 5 times m-reduction [i] based on (15, 70, 216)-net in base 64, using
    - base change [i] based on digital (5, 60, 216)-net over F₁₂₈, using
      - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(146, 146+50, 1655)-Net over F₄ — Digital

Digital (146, 196, 1655)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(146, 146+50, 178075)-Net in Base 4 — Upper bound on s

There is no (146, 196, 178076)-net in base 4, because

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]