MinT - Best Known (172−137, 172, s)-Nets in Base 3

Best Known (172−137, 172, s)-Nets in Base 3

(172−137, 172, 38)-Net over F₃ — Constructive and digital

Digital (35, 172, 38)-net over F₃, using

t-expansion [i] based on digital (32, 172, 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]

(172−137, 172, 47)-Net over F₃ — Digital

Digital (35, 172, 47)-net over F₃, using

net from sequence [i] based on digital (35, 46)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 35 and N(F) ≥ 47, using
  - a curve from the manYPoints database [i]

(172−137, 172, 112)-Net over F₃ — Upper bound on s (digital)

There is no digital (35, 172, 113)-net over F₃, because

65 times m-reduction [i] would yield digital (35, 107, 113)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁰⁷, 113, F₃, 72) (dual of [113, 6, 73]-code), but
  - residual code [i] would yield linear OA(3³⁵, 40, F₃, 24) (dual of [40, 5, 25]-code), but
    - â€œvE1â€ bound on codes from Brouwerâ€™s database [i]

(172−137, 172, 114)-Net in Base 3 — Upper bound on s

There is no (35, 172, 115)-net in base 3, because

70 times m-reduction [i] would yield (35, 102, 115)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹⁰², 115, S₃, 67), but
  - the linear programming bound shows that MÂ â‰¥ 273892 744995 340833 777347 939263 771534 786080 723599 733441 / 56287 > 3¹⁰² [i]