MinT - Best Known (87, 87+67, s)-Nets in Base 4

Best Known (87, 87+67, s)-Nets in Base 4

(87, 87+67, 130)-Net over F₄ — Constructive and digital

Digital (87, 154, 130)-net over F₄, using

8 times m-reduction [i] based on digital (87, 162, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 81, 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]

(87, 87+67, 178)-Net over F₄ — Digital

Digital (87, 154, 178)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(87, 87+67, 2687)-Net in Base 4 — Upper bound on s

There is no (87, 154, 2688)-net in base 4, because

1 times m-reduction [i] would yield (87, 153, 2688)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 131 879920 577670 259889 037627 954552 821505 095567 016688 628238 937662 452172 414096 191554 787581 745125 > 4¹⁵³ [i]