MinT - Best Known (90−66, 90, s)-Nets in Base 4

Best Known (90−66, 90, s)-Nets in Base 4

(90−66, 90, 34)-Net over F₄ — Constructive and digital

Digital (24, 90, 34)-net over F₄, using

t-expansion [i] based on digital (21, 90, 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]

(90−66, 90, 35)-Net in Base 4 — Constructive

(24, 90, 35)-net in base 4, using

net from sequence [i] based on (24, 34)-sequence in base 4, using
- base expansion [i] based on digital (48, 34)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 2 places with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]

(90−66, 90, 49)-Net over F₄ — Digital

Digital (24, 90, 49)-net over F₄, using

net from sequence [i] based on digital (24, 48)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 24 and N(F) ≥ 49, using
  - a curve from the manYPoints database [i]

(90−66, 90, 105)-Net in Base 4 — Upper bound on s

There is no (24, 90, 106)-net in base 4, because

1 times m-reduction [i] would yield (24, 89, 106)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4⁸⁹, 106, S₄, 65), but
  - the linear programming bound shows that MÂ â‰¥ 17537 155141 375370 318427 343564 240274 663299 243711 406953 543164 231680 / 39462 952397 > 4⁸⁹ [i]