MinT - Best Known (138, 138+77, s)-Nets in Base 4

Best Known (138, 138+77, s)-Nets in Base 4

(138, 138+77, 157)-Net over F₄ — Constructive and digital

Digital (138, 215, 157)-net over F₄, using

4¹ times duplication [i] based on digital (137, 214, 157)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 48, 27)-net over F₄, using
    - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
        an explicitly constructive algebraic function field [i]
  2. digital (89, 166, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 83, 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+77, 196)-Net in Base 4 — Constructive

(138, 215, 196)-net in base 4, using

3 times m-reduction [i] based on (138, 218, 196)-net in base 4, using
- trace code for nets [i] based on (29, 109, 98)-net in base 16, using
  - 1 times m-reduction [i] based on (29, 110, 98)-net in base 16, using
    - base change [i] based on digital (7, 88, 98)-net over F₃₂, using
      - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(138, 138+77, 455)-Net over F₄ — Digital

Digital (138, 215, 455)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(138, 138+77, 12278)-Net in Base 4 — Upper bound on s

There is no (138, 215, 12279)-net in base 4, because

1 times m-reduction [i] would yield (138, 214, 12279)-net in base 4, but
- 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]