MinT - Best Known (157−69, 157, s)-Nets in Base 4

Best Known (157−69, 157, s)-Nets in Base 4

(157−69, 157, 130)-Net over F₄ — Constructive and digital

Digital (88, 157, 130)-net over F₄, using

7 times m-reduction [i] based on digital (88, 164, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 82, 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]

(157−69, 157, 175)-Net over F₄ — Digital

Digital (88, 157, 175)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(157−69, 157, 2582)-Net in Base 4 — Upper bound on s

There is no (88, 157, 2583)-net in base 4, because

1 times m-reduction [i] would yield (88, 156, 2583)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8352 343611 133313 646392 810950 882359 655854 643447 561349 603842 972192 039999 033396 666385 691090 313456 > 4¹⁵⁶ [i]