MinT - Best Known (234−175, 234, s)-Nets in Base 4

Best Known (234−175, 234, s)-Nets in Base 4

(234−175, 234, 66)-Net over F₄ — Constructive and digital

Digital (59, 234, 66)-net over F₄, using

t-expansion [i] based on digital (49, 234, 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]

(234−175, 234, 91)-Net over F₄ — Digital

Digital (59, 234, 91)-net over F₄, using

t-expansion [i] based on digital (50, 234, 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]

(234−175, 234, 254)-Net over F₄ — Upper bound on s (digital)

There is no digital (59, 234, 255)-net over F₄, because

3 times m-reduction [i] would yield digital (59, 231, 255)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²³¹, 255, F₄, 172) (dual of [255, 24, 173]-code), but
  - residual code [i] would yield OA(4⁵⁹, 82, S₄, 43), but
    - the linear programming bound shows that MÂ â‰¥ 113 303848 813251 769790 373484 570498 790503 594690 871296 / 335 930280 888125 > 4⁵⁹ [i]

(234−175, 234, 385)-Net in Base 4 — Upper bound on s

There is no (59, 234, 386)-net in base 4, because

1 times m-reduction [i] would yield (59, 233, 386)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 222 551231 337005 992781 063604 410176 840961 294015 725892 128772 349918 310343 751708 715764 251533 630807 360773 862879 712367 150868 904192 463718 941832 270912 > 4²³³ [i]