MinT - Best Known (75−58, 75, s)-Nets in Base 3

Best Known (75−58, 75, s)-Nets in Base 3

(75−58, 75, 28)-Net over F₃ — Constructive and digital

Digital (17, 75, 28)-net over F₃, using

t-expansion [i] based on digital (15, 75, 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]

(75−58, 75, 57)-Net over F₃ — Upper bound on s (digital)

There is no digital (17, 75, 58)-net over F₃, because

22 times m-reduction [i] would yield digital (17, 53, 58)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3⁵³, 58, F₃, 36) (dual of [58, 5, 37]-code), but
  - residual code [i] would yield linear OA(3¹⁷, 21, F₃, 12) (dual of [21, 4, 13]-code), but
    - â€œHNâ€ bound on codes from Brouwerâ€™s database [i]

(75−58, 75, 60)-Net in Base 3 — Upper bound on s

There is no (17, 75, 61)-net in base 3, because

21 times m-reduction [i] would yield (17, 54, 61)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3⁵⁴, 61, S₃, 37), but
  - the linear programming bound shows that MÂ â‰¥ 26690 729284 395387 397889 087571 / 437 > 3⁵⁴ [i]