MinT - Best Known (59, 232, s)-Nets in Base 4

Best Known (59, 232, s)-Nets in Base 4

(59, 232, 66)-Net over F₄ — Constructive and digital

Digital (59, 232, 66)-net over F₄, using

t-expansion [i] based on digital (49, 232, 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]

(59, 232, 91)-Net over F₄ — Digital

Digital (59, 232, 91)-net over F₄, using

t-expansion [i] based on digital (50, 232, 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]

(59, 232, 254)-Net over F₄ — Upper bound on s (digital)

There is no digital (59, 232, 255)-net over F₄, because

1 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]

(59, 232, 385)-Net in Base 4 — Upper bound on s

There is no (59, 232, 386)-net in base 4, because

1 times m-reduction [i] would yield (59, 231, 386)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12 331652 953360 249515 000534 065687 234997 990409 191089 646970 540581 006659 895284 303116 441989 031486 701137 850486 091360 651206 042772 355437 482349 041088 > 4²³¹ [i]