MinT - Best Known (71−48, 71, s)-Nets in Base 4

Best Known (71−48, 71, s)-Nets in Base 4

(71−48, 71, 34)-Net over F₄ — Constructive and digital

Digital (23, 71, 34)-net over F₄, using

t-expansion [i] based on digital (21, 71, 34)-net over F₄, using
- net from sequence [i] based on digital (21, 33)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
    - T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(71−48, 71, 45)-Net over F₄ — Digital

Digital (23, 71, 45)-net over F₄, using

net from sequence [i] based on digital (23, 44)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 23 and N(F) ≥ 45, using
  - a curve from the manYPoints database [i]

(71−48, 71, 174)-Net in Base 4 — Upper bound on s

There is no (23, 71, 175)-net in base 4, because

2 times m-reduction [i] would yield (23, 69, 175)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4⁶⁹, 175, S₄, 46), but
  - the linear programming bound shows that MÂ â‰¥ 672974 508783 713085 919446 392721 434812 374063 221813 152066 788676 290147 045574 983959 198536 005103 820640 027801 847804 723200 000000 / 1 829712 211075 097010 907956 226624 270704 376891 436886 295247 592787 694007 310519 329949 > 4⁶⁹ [i]