MinT - Best Known (22, 67, s)-Nets in Base 3

Best Known (22, 67, s)-Nets in Base 3

(22, 67, 32)-Net over F₃ — Constructive and digital

Digital (22, 67, 32)-net over F₃, using

t-expansion [i] based on digital (21, 67, 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]

(22, 67, 73)-Net over F₃ — Upper bound on s (digital)

There is no digital (22, 67, 74)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3⁶⁷, 74, F₃, 45) (dual of [74, 7, 46]-code), but
- residual code [i] would yield linear OA(3²², 28, F₃, 15) (dual of [28, 6, 16]-code), but
  - â€œHHMâ€ bound on codes from Brouwerâ€™s database [i]

(22, 67, 76)-Net in Base 3 — Upper bound on s

There is no (22, 67, 77)-net in base 3, because

1 times m-reduction [i] would yield (22, 66, 77)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3⁶⁶, 77, S₃, 44), but
  - the linear programming bound shows that MÂ â‰¥ 34 245670 463812 538347 598340 341640 500169 / 1 016275 > 3⁶⁶ [i]