MinT - Best Known (53−35, 53, s)-Nets in Base 3

Best Known (53−35, 53, s)-Nets in Base 3

(53−35, 53, 28)-Net over F₃ — Constructive and digital

Digital (18, 53, 28)-net over F₃, using

t-expansion [i] based on digital (15, 53, 28)-net over F₃, using
- net from sequence [i] based on digital (15, 27)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
    - an explicitly constructive algebraic function field [i]

(53−35, 53, 68)-Net over F₃ — Upper bound on s (digital)

There is no digital (18, 53, 69)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3⁵³, 69, F₃, 35) (dual of [69, 16, 36]-code), but
- â€œLPâ€ bound on codes from Brouwerâ€™s database [i]

(53−35, 53, 69)-Net in Base 3 — Upper bound on s

There is no (18, 53, 70)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3⁵³, 70, S₃, 35), but
- the linear programming bound shows that MÂ â‰¥ 1351 711163 492828 260644 190186 134657 / 50 557336 > 3⁵³ [i]