MinT - Best Known (56, 56+177, s)-Nets in Base 4

Best Known (56, 56+177, s)-Nets in Base 4

(56, 56+177, 66)-Net over F₄ — Constructive and digital

Digital (56, 233, 66)-net over F₄, using

t-expansion [i] based on digital (49, 233, 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+177, 91)-Net over F₄ — Digital

Digital (56, 233, 91)-net over F₄, using

t-expansion [i] based on digital (50, 233, 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+177, 234)-Net over F₄ — Upper bound on s (digital)

There is no digital (56, 233, 235)-net over F₄, because

5 times m-reduction [i] would yield digital (56, 228, 235)-net over F₄, but
- 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+177, 363)-Net in Base 4 — Upper bound on s

There is no (56, 233, 364)-net in base 4, because

1 times m-reduction [i] would yield (56, 232, 364)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 48 716969 316185 581146 234580 323596 291273 006484 452656 136151 822959 539576 996892 660072 185171 479500 832048 452916 249476 395656 759804 805456 778914 303846 > 4²³² [i]