MinT - Best Known (78−55, 78, s)-Nets in Base 3

Best Known (78−55, 78, s)-Nets in Base 3

(78−55, 78, 32)-Net over F₃ — Constructive and digital

Digital (23, 78, 32)-net over F₃, using

t-expansion [i] based on digital (21, 78, 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]

(78−55, 78, 77)-Net over F₃ — Upper bound on s (digital)

There is no digital (23, 78, 78)-net over F₃, because

7 times m-reduction [i] would yield digital (23, 71, 78)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3⁷¹, 78, F₃, 48) (dual of [78, 7, 49]-code), but
  - residual code [i] would yield linear OA(3²³, 29, F₃, 16) (dual of [29, 6, 17]-code), but
    - 1 times truncation [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]

(78−55, 78, 78)-Net in Base 3 — Upper bound on s

There is no (23, 78, 79)-net in base 3, because

7 times m-reduction [i] would yield (23, 71, 79)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3⁷¹, 79, S₃, 48), but
  - the linear programming bound shows that MÂ â‰¥ 31 021606 173381 243165 103883 623561 926657 / 3857 > 3⁷¹ [i]