MinT - Best Known (138−105, 138, s)-Nets in Base 3

Best Known (138−105, 138, s)-Nets in Base 3

(138−105, 138, 38)-Net over F₃ — Constructive and digital

Digital (33, 138, 38)-net over F₃, using

t-expansion [i] based on digital (32, 138, 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]

(138−105, 138, 46)-Net over F₃ — Digital

Digital (33, 138, 46)-net over F₃, using

net from sequence [i] based on digital (33, 45)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 33 and N(F) ≥ 46, using
  - a curve from the manYPoints database [i]

(138−105, 138, 107)-Net over F₃ — Upper bound on s (digital)

There is no digital (33, 138, 108)-net over F₃, because

36 times m-reduction [i] would yield digital (33, 102, 108)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁰², 108, F₃, 69) (dual of [108, 6, 70]-code), but
  - â€œMaâ€ bound on codes from Brouwerâ€™s database [i]

(138−105, 138, 108)-Net in Base 3 — Upper bound on s

There is no (33, 138, 109)-net in base 3, because

41 times m-reduction [i] would yield (33, 97, 109)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3⁹⁷, 109, S₃, 64), but
  - the linear programming bound shows that MÂ â‰¥ 13 468083 991561 019023 891072 124538 296580 900734 840712 818463 / 597 447695 > 3⁹⁷ [i]