MinT - Best Known (178−64, 178, s)-Nets in Base 4

Best Known (178−64, 178, s)-Nets in Base 4

(178−64, 178, 147)-Net over F₄ — Constructive and digital

Digital (114, 178, 147)-net over F₄, using

1 times m-reduction [i] based on digital (114, 179, 147)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 37, 17)-net over F₄, using
    - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
        an explicitly constructive algebraic function field [i]
  2. digital (77, 142, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 71, 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]

(178−64, 178, 196)-Net in Base 4 — Constructive

(114, 178, 196)-net in base 4, using

trace code for nets [i] based on (25, 89, 98)-net in base 16, using
- 1 times m-reduction [i] based on (25, 90, 98)-net in base 16, using
  - base change [i] based on digital (7, 72, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(178−64, 178, 380)-Net over F₄ — Digital

Digital (114, 178, 380)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(178−64, 178, 9495)-Net in Base 4 — Upper bound on s

There is no (114, 178, 9496)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 146964 708987 544446 462676 780667 858427 596593 701706 087637 969161 132951 237104 624835 445379 149902 710026 821081 733896 > 4¹⁷⁸ [i]