MinT - Best Known (24, ∞, s)-Nets in Base 4

Best Known (24, ∞, s)-Nets in Base 4

(24, ∞, 34)-Net over F₄ — Constructive and digital

Digital (24, m, 34)-net over F₄ for arbitrarily large m, using

net from sequence [i] based on digital (24, 33)-sequence over F₄, using
- t-expansion [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]

(24, ∞, 35)-Net in Base 4 — Constructive

(24, m, 35)-net in base 4 for arbitrarily large m, 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]

(24, ∞, 49)-Net over F₄ — Digital

Digital (24, m, 49)-net over F₄ for arbitrarily large m, 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]

(24, ∞, 85)-Net in Base 4 — Upper bound on s

There is no (24, m, 86)-net in base 4 for arbitrarily large m, because

m-reduction [i] would yield (24, 254, 86)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4²⁵⁴, 86, S₄, 3, 230), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 87150 751544 626881 147231 162488 337999 828615 877833 850556 955203 149384 191466 195478 055349 212182 938084 872279 985207 078212 159330 549400 238277 652704 651818 718539 546624 / 77 > 4²⁵⁴ [i]