MinT - Best Known (137−109, 137, s)-Nets in Base 3

Best Known (137−109, 137, s)-Nets in Base 3

(137−109, 137, 37)-Net over F₃ — Constructive and digital

Digital (28, 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]

(137−109, 137, 39)-Net over F₃ — Digital

Digital (28, 137, 39)-net over F₃, using

t-expansion [i] based on digital (27, 137, 39)-net over F₃, using
- net from sequence [i] based on digital (27, 38)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 27 and N(F) ≥ 39, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(137−109, 137, 92)-Net over F₃ — Upper bound on s (digital)

There is no digital (28, 137, 93)-net over F₃, because

49 times m-reduction [i] would yield digital (28, 88, 93)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3⁸⁸, 93, F₃, 60) (dual of [93, 5, 61]-code), but
  - â€œHHâ€ bound on codes from Brouwerâ€™s database [i]

(137−109, 137, 93)-Net in Base 3 — Upper bound on s

There is no (28, 137, 94)-net in base 3, because

51 times m-reduction [i] would yield (28, 86, 94)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3⁸⁶, 94, S₃, 58), but
  - the linear programming bound shows that MÂ â‰¥ 2958 779649 581734 512377 183745 542610 568274 051211 / 23069 > 3⁸⁶ [i]