MinT - Best Known (156−56, 156, s)-Nets in Base 4

Best Known (156−56, 156, s)-Nets in Base 4

(156−56, 156, 145)-Net over F₄ — Constructive and digital

Digital (100, 156, 145)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 32, 15)-net over F₄, using
  - net from sequence [i] based on digital (4, 14)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 4 and N(F) ≥ 15, using
      - an explicitly constructive algebraic function field [i]
2. digital (68, 124, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 62, 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]

(156−56, 156, 152)-Net in Base 4 — Constructive

(100, 156, 152)-net in base 4, using

2 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]

(156−56, 156, 341)-Net over F₄ — Digital

Digital (100, 156, 341)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(156−56, 156, 8492)-Net in Base 4 — Upper bound on s

There is no (100, 156, 8493)-net in base 4, because

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]