MinT - Best Known (138, 138+34, s)-Nets in Base 4

Best Known (138, 138+34, s)-Nets in Base 4

(138, 138+34, 1076)-Net over F₄ — Constructive and digital

Digital (138, 172, 1076)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (19, 36, 48)-net over F₄, using
  - trace code for nets [i] based on digital (1, 18, 24)-net over F₁₆, using
    - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
        a function field by SÃ©mirat [i]
2. digital (102, 136, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 34, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(138, 138+34, 6045)-Net over F₄ — Digital

Digital (138, 172, 6045)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(138, 138+34, 2952925)-Net in Base 4 — Upper bound on s

There is no (138, 172, 2952926)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 35 836085 839795 969436 770916 856691 466764 445376 959857 719667 702531 880058 937053 260428 129612 687360 949324 179549 > 4¹⁷² [i]