MinT - Best Known (157−57, 157, s)-Nets in Base 4

Best Known (157−57, 157, s)-Nets in Base 4

(157−57, 157, 144)-Net over F₄ — Constructive and digital

Digital (100, 157, 144)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 31, 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 (69, 126, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 63, 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]

(157−57, 157, 152)-Net in Base 4 — Constructive

(100, 157, 152)-net in base 4, using

1 times m-reduction [i] based on (100, 158, 152)-net in base 4, using
- trace code for nets [i] based on (21, 79, 76)-net in base 16, using
  - 1 times m-reduction [i] based on (21, 80, 76)-net in base 16, using
    - base change [i] based on digital (5, 64, 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]

(157−57, 157, 330)-Net over F₄ — Digital

Digital (100, 157, 330)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(157−57, 157, 8492)-Net in Base 4 — Upper bound on s

There is no (100, 157, 8493)-net in base 4, because

1 times m-reduction [i] would yield (100, 156, 8493)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8347 972369 765498 294686 332957 446432 120093 740929 879988 354426 414470 867052 902730 425736 687611 112010 > 4¹⁵⁶ [i]