MinT - Best Known (199−144, 199, s)-Nets in Base 4

Best Known (199−144, 199, s)-Nets in Base 4

(199−144, 199, 66)-Net over F₄ — Constructive and digital

Digital (55, 199, 66)-net over F₄, using

t-expansion [i] based on digital (49, 199, 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]

(199−144, 199, 91)-Net over F₄ — Digital

Digital (55, 199, 91)-net over F₄, using

t-expansion [i] based on digital (50, 199, 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]

(199−144, 199, 308)-Net over F₄ — Upper bound on s (digital)

There is no digital (55, 199, 309)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4¹⁹⁹, 309, F₄, 144) (dual of [309, 110, 145]-code), but
- residual code [i] would yield OA(4⁵⁵, 164, S₄, 36), but
  - the linear programming bound shows that MÂ â‰¥ 980 201801 606598 702017 212850 515741 792418 086779 414793 869747 648944 529897 580134 400000 / 752717 700220 343776 849584 966340 992475 812762 611911 > 4⁵⁵ [i]

(199−144, 199, 368)-Net in Base 4 — Upper bound on s

There is no (55, 199, 369)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 706841 696997 614672 671770 925365 513085 627129 255982 925325 816418 625241 390908 171572 003582 306945 686339 953111 295024 653165 825070 > 4¹⁹⁹ [i]