MinT - Best Known (172−69, 172, s)-Nets in Base 4

Best Known (172−69, 172, s)-Nets in Base 4

(172−69, 172, 130)-Net over F₄ — Constructive and digital

Digital (103, 172, 130)-net over F₄, using

22 times m-reduction [i] based on digital (103, 194, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 97, 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]

(172−69, 172, 254)-Net over F₄ — Digital

Digital (103, 172, 254)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−69, 172, 4784)-Net in Base 4 — Upper bound on s

There is no (103, 172, 4785)-net in base 4, because

1 times m-reduction [i] would yield (103, 171, 4785)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8 982494 903127 315670 288617 796328 458061 245377 620762 805868 506853 186130 628716 875798 437869 538276 098429 052184 > 4¹⁷¹ [i]