MinT - Best Known (190−71, 190, s)-Nets in Base 4

Best Known (190−71, 190, s)-Nets in Base 4

(190−71, 190, 139)-Net over F₄ — Constructive and digital

Digital (119, 190, 139)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 36, 9)-net over F₄, using
  - net from sequence [i] based on digital (1, 8)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 1 and N(F) ≥ 9, using
      - the Hermitian function field over F₄ [i]
2. digital (83, 154, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 77, 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]

(190−71, 190, 152)-Net in Base 4 — Constructive

(119, 190, 152)-net in base 4, using

trace code for nets [i] based on (24, 95, 76)-net in base 16, using
- base change [i] based on digital (5, 76, 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]

(190−71, 190, 352)-Net over F₄ — Digital

Digital (119, 190, 352)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(190−71, 190, 8236)-Net in Base 4 — Upper bound on s

There is no (119, 190, 8237)-net in base 4, because

1 times m-reduction [i] would yield (119, 189, 8237)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 616122 786836 102712 376678 401057 415479 259174 363203 505524 875649 011371 910804 021045 757435 125915 850193 561842 868112 311228 > 4¹⁸⁹ [i]