MinT - Best Known (28, 28+∞, s)-Nets in Base 4

Best Known (28, 28+∞, s)-Nets in Base 4

(28, 28+∞, 34)-Net over F₄ — Constructive and digital

Digital (28, m, 34)-net over F₄ for arbitrarily large m, using

net from sequence [i] based on digital (28, 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]

(28, 28+∞, 42)-Net in Base 4 — Constructive

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

net from sequence [i] based on (28, 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]

(28, 28+∞, 55)-Net over F₄ — Digital

Digital (28, m, 55)-net over F₄ for arbitrarily large m, using

net from sequence [i] based on digital (28, 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]

(28, 28+∞, 97)-Net in Base 4 — Upper bound on s

There is no (28, m, 98)-net in base 4 for arbitrarily large m, because

m-reduction [i] would yield (28, 290, 98)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4²⁹⁰, 98, S₄, 3, 262), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 1171 356781 376623 073310 539939 549049 092006 382069 343339 273833 002633 655961 258104 534922 918618 475426 047061 070734 571873 612435 911168 064802 476773 730406 298590 562566 043253 049672 896239 108096 / 263 > 4²⁹⁰ [i]