MinT - Best Known (73−48, 73, s)-Nets in Base 3

Best Known (73−48, 73, s)-Nets in Base 3

(73−48, 73, 32)-Net over F₃ — Constructive and digital

Digital (25, 73, 32)-net over F₃, using

t-expansion [i] based on digital (21, 73, 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]

(73−48, 73, 36)-Net over F₃ — Digital

Digital (25, 73, 36)-net over F₃, using

net from sequence [i] based on digital (25, 35)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 25 and N(F) ≥ 36, using
  - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(73−48, 73, 85)-Net over F₃ — Upper bound on s (digital)

There is no digital (25, 73, 86)-net over F₃, because

2 times m-reduction [i] would yield digital (25, 71, 86)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3⁷¹, 86, F₃, 46) (dual of [86, 15, 47]-code), but
  - â€œDaâ€ bound on codes from Brouwerâ€™s database [i]

(73−48, 73, 87)-Net in Base 3 — Upper bound on s

There is no (25, 73, 88)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3⁷³, 88, S₃, 48), but
- the linear programming bound shows that MÂ â‰¥ 21 779877 362252 895710 497054 495927 589540 336685 / 227 684429 > 3⁷³ [i]