MinT - Best Known (220−61, 220, s)-Nets in Base 4

Best Known (220−61, 220, s)-Nets in Base 4

(220−61, 220, 531)-Net over F₄ — Constructive and digital

Digital (159, 220, 531)-net over F₄, using

8 times m-reduction [i] based on digital (159, 228, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 76, 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]

(220−61, 220, 1263)-Net over F₄ — Digital

Digital (159, 220, 1263)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(220−61, 220, 99677)-Net in Base 4 — Upper bound on s

There is no (159, 220, 99678)-net in base 4, because

1 times m-reduction [i] would yield (159, 219, 99678)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 709837 126480 439651 541004 494109 672529 620252 121176 104581 143346 168943 914324 781866 121441 796167 444024 764187 756349 913635 452368 120288 386840 > 4²¹⁹ [i]