MinT - Best Known (27, 27+72, s)-Nets in Base 4

Best Known (27, 27+72, s)-Nets in Base 4

(27, 27+72, 34)-Net over F₄ — Constructive and digital

Digital (27, 99, 34)-net over F₄, using

t-expansion [i] based on digital (21, 99, 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]

(27, 27+72, 42)-Net in Base 4 — Constructive

(27, 99, 42)-net in base 4, 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, 27+72, 55)-Net over F₄ — Digital

Digital (27, 99, 55)-net over F₄, using

t-expansion [i] based on digital (26, 99, 55)-net over F₄, using
- net from sequence [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, 27+72, 117)-Net in Base 4 — Upper bound on s

There is no (27, 99, 118)-net in base 4, because

1 times m-reduction [i] would yield (27, 98, 118)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4⁹⁸, 118, S₄, 71), but
  - the linear programming bound shows that MÂ â‰¥ 8031 723376 578101 243215 805052 269662 109650 531096 148226 832013 262457 929728 / 66351 579785 > 4⁹⁸ [i]