MinT - Best Known (137−105, 137, s)-Nets in Base 3

Best Known (137−105, 137, s)-Nets in Base 3

(137−105, 137, 38)-Net over F₃ — Constructive and digital

Digital (32, 137, 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]

(137−105, 137, 42)-Net over F₃ — Digital

Digital (32, 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]

(137−105, 137, 104)-Net over F₃ — Upper bound on s (digital)

There is no digital (32, 137, 105)-net over F₃, because

39 times m-reduction [i] would yield digital (32, 98, 105)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3⁹⁸, 105, F₃, 66) (dual of [105, 7, 67]-code), but
  - residual code [i] would yield linear OA(3³², 38, F₃, 22) (dual of [38, 6, 23]-code), but
    - 1 times truncation [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]

(137−105, 137, 105)-Net in Base 3 — Upper bound on s

There is no (32, 137, 106)-net in base 3, because

43 times m-reduction [i] would yield (32, 94, 106)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3⁹⁴, 106, S₃, 62), but
  - the linear programming bound shows that MÂ â‰¥ 36 248218 958151 463616 231099 460182 029513 381562 380737 / 44872 > 3⁹⁴ [i]