MinT - Best Known (84−53, 84, s)-Nets in Base 3

Best Known (84−53, 84, s)-Nets in Base 3

(84−53, 84, 37)-Net over F₃ — Constructive and digital

Digital (31, 84, 37)-net over F₃, using

t-expansion [i] based on digital (27, 84, 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]

(84−53, 84, 42)-Net over F₃ — Digital

Digital (31, 84, 42)-net over F₃, using

t-expansion [i] based on digital (29, 84, 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]

(84−53, 84, 112)-Net in Base 3 — Upper bound on s

There is no (31, 84, 113)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3⁸⁴, 113, S₃, 53), but
- the linear programming bound shows that MÂ â‰¥ 220 344254 536683 114597 471552 228690 132437 887421 343564 925242 / 18336 783241 265797 > 3⁸⁴ [i]