MinT - Best Known (111−89, 111, s)-Nets in Base 4

Best Known (111−89, 111, s)-Nets in Base 4

(111−89, 111, 34)-Net over F₄ — Constructive and digital

Digital (22, 111, 34)-net over F₄, using

t-expansion [i] based on digital (21, 111, 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]

(111−89, 111, 44)-Net over F₄ — Digital

Digital (22, 111, 44)-net over F₄, using

t-expansion [i] based on digital (21, 111, 44)-net over F₄, using
- net from sequence [i] based on digital (21, 43)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 44, using
    - a curve from the manYPoints database [i]

(111−89, 111, 95)-Net over F₄ — Upper bound on s (digital)

There is no digital (22, 111, 96)-net over F₄, because

25 times m-reduction [i] would yield digital (22, 86, 96)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4⁸⁶, 96, F₄, 64) (dual of [96, 10, 65]-code), but
  - â€œGurâ€ bound on codes from Brouwerâ€™s database [i]

(111−89, 111, 96)-Net in Base 4 — Upper bound on s

There is no (22, 111, 97)-net in base 4, because

27 times m-reduction [i] would yield (22, 84, 97)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4⁸⁴, 97, S₄, 62), but
  - the linear programming bound shows that MÂ â‰¥ 1 171133 092329 712265 557448 798148 266278 121440 437172 708782 899200 / 2957 160437 > 4⁸⁴ [i]