MinT - Best Known (237−59, 237, s)-Nets in Base 4

Best Known (237−59, 237, s)-Nets in Base 4

(237−59, 237, 1028)-Net over F₄ — Constructive and digital

Digital (178, 237, 1028)-net over F₄, using

4¹ times duplication [i] based on digital (177, 236, 1028)-net over F₄, using
- trace code for nets [i] based on digital (0, 59, 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]

(237−59, 237, 2188)-Net over F₄ — Digital

Digital (178, 237, 2188)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(237−59, 237, 308666)-Net in Base 4 — Upper bound on s

There is no (178, 237, 308667)-net in base 4, because

1 times m-reduction [i] would yield (178, 236, 308667)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12194 359664 852213 177924 967390 357060 639642 889917 876485 506593 914227 099210 102749 096721 709665 414781 241202 667676 959224 438088 032606 416576 330798 253168 > 4²³⁶ [i]