MinT - Best Known (221−55, 221, s)-Nets in Base 4

Best Known (221−55, 221, s)-Nets in Base 4

(221−55, 221, 1028)-Net over F₄ — Constructive and digital

Digital (166, 221, 1028)-net over F₄, using

4¹ times duplication [i] based on digital (165, 220, 1028)-net over F₄, using
- trace code for nets [i] based on digital (0, 55, 257)-net over F₂₅₆, using
  - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
      - the rational function field F₂₅₆(x) [i]
    - Niederreiter sequence [i]

(221−55, 221, 2061)-Net over F₄ — Digital

Digital (166, 221, 2061)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(221−55, 221, 293040)-Net in Base 4 — Upper bound on s

There is no (166, 221, 293041)-net in base 4, because

1 times m-reduction [i] would yield (166, 220, 293041)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 839473 495066 875765 468187 356967 986026 153354 707236 292420 964550 298711 058752 017362 989595 027182 974302 237178 372826 454644 289290 134468 294752 > 4²²⁰ [i]