MinT - Best Known (138, 138+76, s)-Nets in Base 4

Best Known (138, 138+76, s)-Nets in Base 4

(138, 138+76, 158)-Net over F₄ — Constructive and digital

Digital (138, 214, 158)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (12, 50, 28)-net over F₄, using
  - net from sequence [i] based on digital (12, 27)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 12 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]
2. digital (88, 164, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 82, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(138, 138+76, 208)-Net in Base 4 — Constructive

(138, 214, 208)-net in base 4, using

trace code for nets [i] based on (31, 107, 104)-net in base 16, using
- 3 times m-reduction [i] based on (31, 110, 104)-net in base 16, using
  - base change [i] based on digital (9, 88, 104)-net over F₃₂, using
    - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(138, 138+76, 466)-Net over F₄ — Digital

Digital (138, 214, 466)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(138, 138+76, 12278)-Net in Base 4 — Upper bound on s

There is no (138, 214, 12279)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 695 146177 969944 146116 332689 211126 083601 452019 827252 266819 940073 356571 794403 216061 971485 168205 938457 780388 963340 037962 041539 109009 > 4²¹⁴ [i]