MinT - Best Known (139, 139+50, s)-Nets in Base 4

Best Known (139, 139+50, s)-Nets in Base 4

(139, 139+50, 531)-Net over F₄ — Constructive and digital

Digital (139, 189, 531)-net over F₄, using

9 times m-reduction [i] based on digital (139, 198, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 66, 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]

(139, 139+50, 576)-Net in Base 4 — Constructive

(139, 189, 576)-net in base 4, using

t-expansion [i] based on (138, 189, 576)-net in base 4, using
- trace code for nets [i] based on (12, 63, 192)-net in base 64, using
  - base change [i] based on digital (3, 54, 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]

(139, 139+50, 1362)-Net over F₄ — Digital

Digital (139, 189, 1362)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(139, 139+50, 120782)-Net in Base 4 — Upper bound on s

There is no (139, 189, 120783)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 615684 924502 168649 499214 061313 223382 452456 933836 344953 958234 060096 111417 001820 970193 189510 075380 248096 449743 010712 > 4¹⁸⁹ [i]