MinT - Best Known (209−77, 209, s)-Nets in Base 4

Best Known (209−77, 209, s)-Nets in Base 4

(209−77, 209, 147)-Net over F₄ — Constructive and digital

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]

(209−77, 209, 152)-Net in Base 4 — Constructive

(132, 209, 152)-net in base 4, using

t-expansion [i] based on (131, 209, 152)-net in base 4, using
- 1 times m-reduction [i] based on (131, 210, 152)-net in base 4, using
  - trace code for nets [i] based on (26, 105, 76)-net in base 16, using
    - base change [i] based on digital (5, 84, 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]

(209−77, 209, 402)-Net over F₄ — Digital

Digital (132, 209, 402)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(209−77, 209, 9858)-Net in Base 4 — Upper bound on s

There is no (132, 209, 9859)-net in base 4, because

1 times m-reduction [i] would yield (132, 208, 9859)-net in base 4, but
- 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]