MinT - Best Known (27, 27+189, s)-Nets in Base 4

Best Known (27, 27+189, s)-Nets in Base 4

(27, 27+189, 34)-Net over F₄ — Constructive and digital

Digital (27, 216, 34)-net over F₄, using

t-expansion [i] based on digital (21, 216, 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+189, 42)-Net in Base 4 — Constructive

(27, 216, 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+189, 55)-Net over F₄ — Digital

Digital (27, 216, 55)-net over F₄, using

t-expansion [i] based on digital (26, 216, 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+189, 98)-Net in Base 4 — Upper bound on s

There is no (27, 216, 99)-net in base 4, because

22 times m-reduction [i] would yield (27, 194, 99)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4¹⁹⁴, 99, S₄, 2, 167), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 6304 320991 423116 673964 646416 022978 208812 758283 274471 466871 726944 679315 483439 553697 826282 600781 586502 529060 478449 090560 / 7 > 4¹⁹⁴ [i]