MinT - Best Known (166−53, 166, s)-Nets in Base 4

Best Known (166−53, 166, s)-Nets in Base 4

(166−53, 166, 240)-Net over F₄ — Constructive and digital

Digital (113, 166, 240)-net over F₄, using

2 times m-reduction [i] based on digital (113, 168, 240)-net over F₄, using
- trace code for nets [i] based on digital (1, 56, 80)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
      - a function field by SÃ©mirat [i]

(166−53, 166, 553)-Net over F₄ — Digital

Digital (113, 166, 553)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(166−53, 166, 23256)-Net in Base 4 — Upper bound on s

There is no (113, 166, 23257)-net in base 4, because

1 times m-reduction [i] would yield (113, 165, 23257)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2188 759821 487451 399698 623865 018087 925029 073432 074111 401855 756291 519277 248116 823382 017682 598540 871088 > 4¹⁶⁵ [i]