MinT - Best Known (166−49, 166, s)-Nets in Base 4

Best Known (166−49, 166, s)-Nets in Base 4

(166−49, 166, 384)-Net over F₄ — Constructive and digital

Digital (117, 166, 384)-net over F₄, using

2 times m-reduction [i] based on digital (117, 168, 384)-net over F₄, using
- trace code for nets [i] based on digital (5, 56, 128)-net over F₆₄, using
  - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
      - a function field by SÃ©mirat [i]

(166−49, 166, 762)-Net over F₄ — Digital

Digital (117, 166, 762)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(166−49, 166, 45000)-Net in Base 4 — Upper bound on s

There is no (117, 166, 45001)-net in base 4, because

1 times m-reduction [i] would yield (117, 165, 45001)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2188 019685 368089 001824 675170 958309 378342 838017 323818 851392 469360 109612 793445 243651 932773 199835 612156 > 4¹⁶⁵ [i]