MinT - Best Known (123, 123+70, s)-Nets in Base 4

Best Known (123, 123+70, s)-Nets in Base 4

(123, 123+70, 147)-Net over F₄ — Constructive and digital

Digital (123, 193, 147)-net over F₄, using

1 times m-reduction [i] based on digital (123, 194, 147)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 40, 17)-net over F₄, using
    - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
        an explicitly constructive algebraic function field [i]
  2. digital (83, 154, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 77, 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]

(123, 123+70, 152)-Net in Base 4 — Constructive

(123, 193, 152)-net in base 4, using

3 times m-reduction [i] based on (123, 196, 152)-net in base 4, using
- trace code for nets [i] based on (25, 98, 76)-net in base 16, using
  - 2 times m-reduction [i] based on (25, 100, 76)-net in base 16, using
    - base change [i] based on digital (5, 80, 76)-net over F₃₂, using
      - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
        a function field by SÃ©mirat [i]

(123, 123+70, 396)-Net over F₄ — Digital

Digital (123, 193, 396)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(123, 123+70, 9655)-Net in Base 4 — Upper bound on s

There is no (123, 193, 9656)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 157 733498 980333 303118 174099 403474 325423 221893 163207 784753 948012 798026 990181 725866 418572 173702 375059 523156 922714 981695 > 4¹⁹³ [i]