MinT - Best Known (82−59, 82, s)-Nets in Base 4

Best Known (82−59, 82, s)-Nets in Base 4

(82−59, 82, 34)-Net over F₄ — Constructive and digital

Digital (23, 82, 34)-net over F₄, using

t-expansion [i] based on digital (21, 82, 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]

(82−59, 82, 45)-Net over F₄ — Digital

Digital (23, 82, 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]

(82−59, 82, 108)-Net in Base 4 — Upper bound on s

There is no (23, 82, 109)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁸², 109, S₄, 59), but
- the linear programming bound shows that MÂ â‰¥ 2066 843532 090158 030067 509856 496495 431763 432665 749400 796952 813922 942976 / 82 680097 196282 961375 > 4⁸² [i]