MinT - Best Known (31, 137, s)-Nets in Base 3

Best Known (31, 137, s)-Nets in Base 3

(31, 137, 37)-Net over F₃ — Constructive and digital

Digital (31, 137, 37)-net over F₃, using

t-expansion [i] based on digital (27, 137, 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]

(31, 137, 42)-Net over F₃ — Digital

Digital (31, 137, 42)-net over F₃, using

t-expansion [i] based on digital (29, 137, 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]

(31, 137, 100)-Net over F₃ — Upper bound on s (digital)

There is no digital (31, 137, 101)-net over F₃, because

43 times m-reduction [i] would yield digital (31, 94, 101)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3⁹⁴, 101, F₃, 63) (dual of [101, 7, 64]-code), but
  - residual code [i] would yield linear OA(3³¹, 37, F₃, 21) (dual of [37, 6, 22]-code), but
    - â€œBouâ€ bound on codes from Brouwerâ€™s database [i]

(31, 137, 102)-Net in Base 3 — Upper bound on s

There is no (31, 137, 103)-net in base 3, because

46 times m-reduction [i] would yield (31, 91, 103)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3⁹¹, 103, S₃, 60), but
  - the linear programming bound shows that MÂ â‰¥ 4 638397 686588 101979 328150 167890 591454 318967 698009 / 161161 > 3⁹¹ [i]