MinT - Best Known (52−35, 52, s)-Nets in Base 3

Best Known (52−35, 52, s)-Nets in Base 3

(52−35, 52, 28)-Net over F₃ — Constructive and digital

Digital (17, 52, 28)-net over F₃, using

t-expansion [i] based on digital (15, 52, 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]

(52−35, 52, 61)-Net over F₃ — Upper bound on s (digital)

There is no digital (17, 52, 62)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3⁵², 62, F₃, 35) (dual of [62, 10, 36]-code), but
- â€œGurâ€ bound on codes from Brouwerâ€™s database [i]

(52−35, 52, 62)-Net in Base 3 — Upper bound on s

There is no (17, 52, 63)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3⁵², 63, S₃, 35), but
- the linear programming bound shows that MÂ â‰¥ 36110 986678 887877 067732 294949 / 4214 > 3⁵² [i]