MinT - Best Known (216−80, 216, s)-Nets in Base 4

Best Known (216−80, 216, s)-Nets in Base 4

(216−80, 216, 145)-Net over F₄ — Constructive and digital

Digital (136, 216, 145)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 44, 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 (92, 172, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 86, 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]

(216−80, 216, 152)-Net in Base 4 — Constructive

(136, 216, 152)-net in base 4, using

2 times m-reduction [i] based on (136, 218, 152)-net in base 4, using
- trace code for nets [i] based on (27, 109, 76)-net in base 16, using
  - 1 times m-reduction [i] based on (27, 110, 76)-net in base 16, using
    - base change [i] based on digital (5, 88, 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]

(216−80, 216, 406)-Net over F₄ — Digital

Digital (136, 216, 406)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(216−80, 216, 9338)-Net in Base 4 — Upper bound on s

There is no (136, 216, 9339)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11120 025790 951101 372726 207462 557672 551618 611481 222079 904083 261519 194625 396347 314059 880455 886895 891474 898751 014817 039045 867680 506075 > 4²¹⁶ [i]