MinT - Best Known (29, ∞, s)-Nets in Base 4

Best Known (29, ∞, s)-Nets in Base 4

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

Digital (29, m, 34)-net over F₄ for arbitrarily large m, using

net from sequence [i] based on digital (29, 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]

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

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

net from sequence [i] based on (29, 41)-sequence in base 4, using
- t-expansion [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]

(29, ∞, 55)-Net over F₄ — Digital

Digital (29, m, 55)-net over F₄ for arbitrarily large m, using

net from sequence [i] based on digital (29, 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]

(29, ∞, 100)-Net in Base 4 — Upper bound on s

There is no (29, m, 101)-net in base 4 for arbitrarily large m, because

m-reduction [i] would yield (29, 299, 101)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4²⁹⁹, 101, S₄, 3, 270), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 302 914636 528312 485971 405774 049454 764023 908574 953697 893699 682985 067417 655233 142294 687178 614573 138576 983038 445722 624957 610314 846028 505059 597475 741862 890319 349155 466271 648310 195229 032448 / 271 > 4²⁹⁹ [i]