MinT - Best Known (71−47, 71, s)-Nets in Base 3

Best Known (71−47, 71, s)-Nets in Base 3

(71−47, 71, 32)-Net over F₃ — Constructive and digital

Digital (24, 71, 32)-net over F₃, using

t-expansion [i] based on digital (21, 71, 32)-net over F₃, using
- net from sequence [i] based on digital (21, 31)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 21 and N(F) ≥ 32, using
    - an explicitly constructive algebraic function field [i]

(71−47, 71, 83)-Net over F₃ — Upper bound on s (digital)

There is no digital (24, 71, 84)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3⁷¹, 84, F₃, 47) (dual of [84, 13, 48]-code), but
- â€œLPâ€ bound on codes from Brouwerâ€™s database [i]

(71−47, 71, 84)-Net in Base 3 — Upper bound on s

There is no (24, 71, 85)-net in base 3, because

1 times m-reduction [i] would yield (24, 70, 85)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3⁷⁰, 85, S₃, 46), but
  - the linear programming bound shows that MÂ â‰¥ 640 504927 462165 667703 027244 227660 956271 / 192089 > 3⁷⁰ [i]