MinT - Best Known (27, ∞, s)-Nets in Base 4

Best Known (27, ∞, s)-Nets in Base 4

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

Digital (27, m, 34)-net over F₄ for arbitrarily large m, using

net from sequence [i] based on digital (27, 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]

(27, ∞, 42)-Net in Base 4 — Constructive

(27, m, 42)-net in base 4 for arbitrarily large m, using

net from sequence [i] based on (27, 41)-sequence in base 4, using
- base expansion [i] based on digital (54, 41)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
    - an explicitly constructive algebraic function field [i]

(27, ∞, 55)-Net over F₄ — Digital

Digital (27, m, 55)-net over F₄ for arbitrarily large m, using

net from sequence [i] based on digital (27, 54)-sequence over F₄, using
- t-expansion [i] based on digital (26, 54)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 26 and N(F) ≥ 55, using
    - Drinfeld modules of rank 1 [i]

(27, ∞, 94)-Net in Base 4 — Upper bound on s

There is no (27, m, 95)-net in base 4 for arbitrarily large m, because

m-reduction [i] would yield (27, 281, 95)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4²⁸¹, 95, S₄, 3, 254), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 301 916993 985723 308179 324366 479061 511273 353697 633315 234270 096504 019649 932991 371908 166890 138014 212701 403317 470002 461107 591981 646770 393983 410604 914740 114615 683491 951626 158080 / 17 > 4²⁸¹ [i]