MinT - Best Known (226−169, 226, s)-Nets in Base 4

Best Known (226−169, 226, s)-Nets in Base 4

(226−169, 226, 66)-Net over F₄ — Constructive and digital

Digital (57, 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]

(226−169, 226, 91)-Net over F₄ — Digital

Digital (57, 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]

(226−169, 226, 243)-Net over F₄ — Upper bound on s (digital)

There is no digital (57, 226, 244)-net over F₄, because

1 times m-reduction [i] would yield digital (57, 225, 244)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²²⁵, 244, F₄, 168) (dual of [244, 19, 169]-code), but
  - residual code [i] would yield OA(4⁵⁷, 75, S₄, 42), but
    - the linear programming bound shows that MÂ â‰¥ 175541 891167 451460 543215 010075 896682 965259 780096 / 6 746458 181029 > 4⁵⁷ [i]

(226−169, 226, 372)-Net in Base 4 — Upper bound on s

There is no (57, 226, 373)-net in base 4, because

1 times m-reduction [i] would yield (57, 225, 373)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3111 245104 181555 954482 433028 419001 009900 332772 208947 261369 948732 275408 753206 643016 450994 453271 139356 299886 719814 745785 054940 277727 922781 > 4²²⁵ [i]