MinT - Best Known (97−68, 97, s)-Nets in Base 3

Best Known (97−68, 97, s)-Nets in Base 3

(97−68, 97, 37)-Net over F₃ — Constructive and digital

Digital (29, 97, 37)-net over F₃, using

t-expansion [i] based on digital (27, 97, 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]

(97−68, 97, 42)-Net over F₃ — Digital

Digital (29, 97, 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]

(97−68, 97, 95)-Net over F₃ — Upper bound on s (digital)

There is no digital (29, 97, 96)-net over F₃, because

8 times m-reduction [i] would yield digital (29, 89, 96)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3⁸⁹, 96, F₃, 60) (dual of [96, 7, 61]-code), but
  - residual code [i] would yield linear OA(3²⁹, 35, F₃, 20) (dual of [35, 6, 21]-code), but
    - â€œBouâ€ bound on codes from Brouwerâ€™s database [i]

(97−68, 97, 96)-Net in Base 3 — Upper bound on s

There is no (29, 97, 97)-net in base 3, because

8 times m-reduction [i] would yield (29, 89, 97)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3⁸⁹, 97, S₃, 60), but
  - the linear programming bound shows that MÂ â‰¥ 532344 681908 373843 992394 006465 724030 651148 098857 / 151585 > 3⁸⁹ [i]