MinT - Best Known (142, 142+85, s)-Nets in Base 4

Best Known (142, 142+85, s)-Nets in Base 4

(142, 142+85, 144)-Net over F₄ — Constructive and digital

Digital (142, 227, 144)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 45, 14)-net over F₄, using
  - net from sequence [i] based on digital (3, 13)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, using
      - an explicitly constructive algebraic function field [i]
2. digital (97, 182, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 91, 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]

(142, 142+85, 152)-Net in Base 4 — Constructive

(142, 227, 152)-net in base 4, using

1 times m-reduction [i] based on (142, 228, 152)-net in base 4, using
- trace code for nets [i] based on (28, 114, 76)-net in base 16, using
  - 1 times m-reduction [i] based on (28, 115, 76)-net in base 16, using
    - base change [i] based on digital (5, 92, 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]

(142, 142+85, 408)-Net over F₄ — Digital

Digital (142, 227, 408)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(142, 142+85, 9522)-Net in Base 4 — Upper bound on s

There is no (142, 227, 9523)-net in base 4, because

1 times m-reduction [i] would yield (142, 226, 9523)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11635 484156 201979 311950 071341 411197 607729 384808 139199 221922 918914 015861 769569 109625 752224 826684 233998 074300 958302 067532 014478 324630 268122 > 4²²⁶ [i]