MinT - Best Known (56, 56+172, s)-Nets in Base 4

Best Known (56, 56+172, s)-Nets in Base 4

(56, 56+172, 66)-Net over F₄ — Constructive and digital

Digital (56, 228, 66)-net over F₄, using

t-expansion [i] based on digital (49, 228, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(56, 56+172, 91)-Net over F₄ — Digital

Digital (56, 228, 91)-net over F₄, using

t-expansion [i] based on digital (50, 228, 91)-net over F₄, using
- net from sequence [i] based on digital (50, 90)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 50 and N(F) ≥ 91, using
    - a curve from the manYPoints database [i]

(56, 56+172, 234)-Net over F₄ — Upper bound on s (digital)

There is no digital (56, 228, 235)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²²⁸, 235, F₄, 172) (dual of [235, 7, 173]-code), but
- residual code [i] would yield OA(4⁵⁶, 62, S₄, 43), but
  - the linear programming bound shows that MÂ â‰¥ 26 916866 914644 546426 302092 970676 977664 / 4675 > 4⁵⁶ [i]

(56, 56+172, 364)-Net in Base 4 — Upper bound on s

There is no (56, 228, 365)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 201191 304874 291089 082986 664777 470188 452092 872351 191477 499226 620193 947684 867230 847645 650453 851623 379038 069359 340220 831275 763424 050971 064088 > 4²²⁸ [i]