MinT - Best Known (251−75, 251, s)-Nets in Base 4

Best Known (251−75, 251, s)-Nets in Base 4

(251−75, 251, 531)-Net over F₄ — Constructive and digital

Digital (176, 251, 531)-net over F₄, using

t-expansion [i] based on digital (175, 251, 531)-net over F₄, using
- 1 times m-reduction [i] based on digital (175, 252, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 84, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(251−75, 251, 1025)-Net over F₄ — Digital

Digital (176, 251, 1025)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(251−75, 251, 57086)-Net in Base 4 — Upper bound on s

There is no (176, 251, 57087)-net in base 4, because

1 times m-reduction [i] would yield (176, 250, 57087)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 274620 439473 063897 815234 074883 374901 869768 396567 665434 777455 014209 264831 969189 878755 210375 407617 942348 708394 581487 704678 510694 784156 894164 169731 031490 > 4²⁵⁰ [i]