MinT - Best Known (226−168, 226, s)-Nets in Base 4

Best Known (226−168, 226, s)-Nets in Base 4

(226−168, 226, 66)-Net over F₄ — Constructive and digital

Digital (58, 226, 66)-net over F₄, using

t-expansion [i] based on digital (49, 226, 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]

(226−168, 226, 91)-Net over F₄ — Digital

Digital (58, 226, 91)-net over F₄, using

t-expansion [i] based on digital (50, 226, 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]

(226−168, 226, 256)-Net over F₄ — Upper bound on s (digital)

There is no digital (58, 226, 257)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²²⁶, 257, F₄, 168) (dual of [257, 31, 169]-code), but
- residual code [i] would yield OA(4⁵⁸, 88, S₄, 42), but
  - the linear programming bound shows that MÂ â‰¥ 158 237175 107336 529531 271038 987459 138072 539269 830656 314178 535424 / 1500 577564 280935 407980 323611 > 4⁵⁸ [i]

(226−168, 226, 379)-Net in Base 4 — Upper bound on s

There is no (58, 226, 380)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11891 088293 973549 957063 410612 988340 431472 045074 499920 492982 363981 012073 826849 802201 015010 100529 746921 482915 355742 758231 962739 185970 446206 > 4²²⁶ [i]