MinT - Best Known (32, 32+212, s)-Nets in Base 3

Best Known (32, 32+212, s)-Nets in Base 3

(32, 32+212, 38)-Net over F₃ — Constructive and digital

Digital (32, 244, 38)-net over F₃, using

net from sequence [i] based on digital (32, 37)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 32 and N(F) ≥ 38, using
  - an explicitly constructive algebraic function field [i]

(32, 32+212, 42)-Net over F₃ — Digital

Digital (32, 244, 42)-net over F₃, using

t-expansion [i] based on digital (29, 244, 42)-net over F₃, using
- net from sequence [i] based on digital (29, 41)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 29 and N(F) ≥ 42, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(32, 32+212, 76)-Net in Base 3 — Upper bound on s

There is no (32, 244, 77)-net in base 3, because

18 times m-reduction [i] would yield (32, 226, 77)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3²²⁶, 77, S₃, 3, 194), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 58 738495 600927 833455 292008 118108 749310 126504 170601 536699 629952 388283 838058 605666 117072 911251 431178 936647 476623 / 65 > 3²²⁶ [i]