MinT - Best Known (212−182, 212, s)-Nets in Base 3

Best Known (212−182, 212, s)-Nets in Base 3

(212−182, 212, 37)-Net over F₃ — Constructive and digital

Digital (30, 212, 37)-net over F₃, using

t-expansion [i] based on digital (27, 212, 37)-net over F₃, using
- net from sequence [i] based on digital (27, 36)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 26, N(F)Â =Â 36, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using an explicitly constructive algebraic function field [i]

(212−182, 212, 42)-Net over F₃ — Digital

Digital (30, 212, 42)-net over F₃, using

t-expansion [i] based on digital (29, 212, 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]

(212−182, 212, 77)-Net in Base 3 — Upper bound on s

There is no (30, 212, 78)-net in base 3, because

61 times m-reduction [i] would yield (30, 151, 78)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3¹⁵¹, 78, S₃, 2, 121), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 89 907201 863535 854420 702290 135762 284537 312963 394702 682637 089810 488324 824507 / 61 > 3¹⁵¹ [i]