MinT - Best Known (186−67, 186, s)-Nets in Base 4

Best Known (186−67, 186, s)-Nets in Base 4

(186−67, 186, 151)-Net over F₄ — Constructive and digital

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]

(186−67, 186, 196)-Net in Base 4 — Constructive

(119, 186, 196)-net in base 4, using

trace code for nets [i] based on (26, 93, 98)-net in base 16, using
- 2 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]

(186−67, 186, 393)-Net over F₄ — Digital

Digital (119, 186, 393)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(186−67, 186, 10382)-Net in Base 4 — Upper bound on s

There is no (119, 186, 10383)-net in base 4, because

1 times m-reduction [i] would yield (119, 185, 10383)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2410 753542 544261 465689 886248 988821 443742 827261 036251 409320 670331 026711 166325 308866 944230 051204 533462 406353 377960 > 4¹⁸⁵ [i]