MinT - Best Known (208−76, 208, s)-Nets in Base 4

Best Known (208−76, 208, s)-Nets in Base 4

(208−76, 208, 147)-Net over F₄ — Constructive and digital

Digital (132, 208, 147)-net over F₄, using

1 times m-reduction [i] based on digital (132, 209, 147)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 43, 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 (89, 166, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 83, 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]

(208−76, 208, 196)-Net in Base 4 — Constructive

(132, 208, 196)-net in base 4, using

trace code for nets [i] based on (28, 104, 98)-net in base 16, using
- 1 times m-reduction [i] based on (28, 105, 98)-net in base 16, using
  - base change [i] based on digital (7, 84, 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]

(208−76, 208, 412)-Net over F₄ — Digital

Digital (132, 208, 412)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(208−76, 208, 9858)-Net in Base 4 — Upper bound on s

There is no (132, 208, 9859)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 169731 186503 901171 492155 996629 689730 159501 061501 635932 994349 130478 628745 306469 182886 037624 430080 808631 992115 961683 516483 363360 > 4²⁰⁸ [i]