MinT - Best Known (187−67, 187, s)-Nets in Base 4

Best Known (187−67, 187, s)-Nets in Base 4

(187−67, 187, 151)-Net over F₄ — Constructive and digital

Digital (120, 187, 151)-net over F₄, using

4¹ times duplication [i] based on digital (119, 186, 151)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 40, 21)-net over F₄, using
    - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, using
        an explicitly constructive algebraic function field [i]
  2. digital (79, 146, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 73, 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]

(187−67, 187, 196)-Net in Base 4 — Constructive

(120, 187, 196)-net in base 4, using

1 times m-reduction [i] based on (120, 188, 196)-net in base 4, using
- trace code for nets [i] based on (26, 94, 98)-net in base 16, using
  - 1 times m-reduction [i] based on (26, 95, 98)-net in base 16, using
    - base change [i] based on digital (7, 76, 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]

(187−67, 187, 402)-Net over F₄ — Digital

Digital (120, 187, 402)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(187−67, 187, 10828)-Net in Base 4 — Upper bound on s

There is no (120, 187, 10829)-net in base 4, because

1 times m-reduction [i] would yield (120, 186, 10829)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 9624 679076 345263 987723 599476 280807 733945 287576 407578 107762 387232 710564 752921 383193 059410 164886 485397 443576 531824 > 4¹⁸⁶ [i]