MinT - Best Known (230−80, 230, s)-Nets in Base 4

Best Known (230−80, 230, s)-Nets in Base 4

(230−80, 230, 163)-Net over F₄ — Constructive and digital

Digital (150, 230, 163)-net over F₄, using

2 times m-reduction [i] based on digital (150, 232, 163)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 56, 33)-net over F₄, using
    - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
  2. digital (94, 176, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 88, 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]

(230−80, 230, 240)-Net in Base 4 — Constructive

(150, 230, 240)-net in base 4, using

t-expansion [i] based on (149, 230, 240)-net in base 4, using
- trace code for nets [i] based on (34, 115, 120)-net in base 16, using
  - base change [i] based on digital (11, 92, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]

(230−80, 230, 534)-Net over F₄ — Digital

Digital (150, 230, 534)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(230−80, 230, 15190)-Net in Base 4 — Upper bound on s

There is no (150, 230, 15191)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 979442 503257 652560 705967 306286 181178 066105 133136 031149 894569 220644 907054 682368 927219 203677 661346 699960 699825 661259 391336 774629 626452 387260 > 4²³⁰ [i]