MinT - Best Known (193−170, 193, s)-Nets in Base 3

Best Known (193−170, 193, s)-Nets in Base 3

(193−170, 193, 32)-Net over F₃ — Constructive and digital

Digital (23, 193, 32)-net over F₃, using

t-expansion [i] based on digital (21, 193, 32)-net over F₃, using
- net from sequence [i] based on digital (21, 31)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 21 and N(F) ≥ 32, using
    - an explicitly constructive algebraic function field [i]

(193−170, 193, 57)-Net in Base 3 — Upper bound on s

There is no (23, 193, 58)-net in base 3, because

24 times m-reduction [i] would yield (23, 169, 58)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3¹⁶⁹, 58, S₃, 3, 146), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 32251 751954 252566 656204 900896 126759 873336 043541 124827 145252 911033 614937 316056 596225 / 49 > 3¹⁶⁹ [i]