MinT - Best Known (81−72, 81, s)-Nets in Base 4

Best Known (81−72, 81, s)-Nets in Base 4

(81−72, 81, 22)-Net over F₄ — Constructive and digital

Digital (9, 81, 22)-net over F₄, using

net from sequence [i] based on digital (9, 21)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 9 and N(F) ≥ 22, using
  - an explicitly constructive algebraic function field [i]

(81−72, 81, 26)-Net over F₄ — Digital

Digital (9, 81, 26)-net over F₄, using

net from sequence [i] based on digital (9, 25)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 9 and N(F) ≥ 26, using
  - Drinfeld modules of rank 1 [i]

(81−72, 81, 38)-Net in Base 4 — Upper bound on s

There is no (9, 81, 39)-net in base 4, because

6 times m-reduction [i] would yield (9, 75, 39)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4⁷⁵, 39, S₄, 2, 66), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 108470 824645 652950 960429 733678 161630 365088 743424 / 67 > 4⁷⁵ [i]