MinT - Best Known (204−148, 204, s)-Nets in Base 4

Best Known (204−148, 204, s)-Nets in Base 4

(204−148, 204, 66)-Net over F₄ — Constructive and digital

Digital (56, 204, 66)-net over F₄, using

t-expansion [i] based on digital (49, 204, 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]

(204−148, 204, 91)-Net over F₄ — Digital

Digital (56, 204, 91)-net over F₄, using

t-expansion [i] based on digital (50, 204, 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]

(204−148, 204, 307)-Net over F₄ — Upper bound on s (digital)

There is no digital (56, 204, 308)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²⁰⁴, 308, F₄, 148) (dual of [308, 104, 149]-code), but
- residual code [i] would yield OA(4⁵⁶, 159, S₄, 37), but
  - the linear programming bound shows that MÂ â‰¥ 552280 967540 130556 225380 165062 844509 113927 435157 299702 522905 668351 970980 659200 000000 / 99 260712 948022 423958 535222 844990 439800 493623 078423 > 4⁵⁶ [i]

(204−148, 204, 374)-Net in Base 4 — Upper bound on s

There is no (56, 204, 375)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 784 325320 317735 177865 390867 594640 517604 026027 108493 398166 295796 178625 505379 247024 099777 331723 758523 271596 144066 696456 844846 > 4²⁰⁴ [i]