MinT - Best Known (32, 32+51, s)-Nets in Base 3

Best Known (32, 32+51, s)-Nets in Base 3

(32, 32+51, 38)-Net over F₃ — Constructive and digital

Digital (32, 83, 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+51, 42)-Net over F₃ — Digital

Digital (32, 83, 42)-net over F₃, using

t-expansion [i] based on digital (29, 83, 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+51, 133)-Net in Base 3 — Upper bound on s

There is no (32, 83, 134)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3⁸³, 134, S₃, 51), but
- the linear programming bound shows that MÂ â‰¥ 625760 414962 993510 058854 039475 945995 304619 276439 915262 016003 058873 437945 073915 147163 250093 / 135 349071 693745 981183 738057 225151 833299 159482 850560 > 3⁸³ [i]