MinT - Best Known (49, 49+109, s)-Nets in Base 3

Best Known (49, 49+109, s)-Nets in Base 3

(49, 49+109, 48)-Net over F₃ — Constructive and digital

Digital (49, 158, 48)-net over F₃, using

t-expansion [i] based on digital (45, 158, 48)-net over F₃, using
- net from sequence [i] based on digital (45, 47)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
    - an explicitly constructive algebraic function field [i]

(49, 49+109, 64)-Net over F₃ — Digital

Digital (49, 158, 64)-net over F₃, using

net from sequence [i] based on digital (49, 63)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 49 and N(F) ≥ 64, using
  - a curve from the manYPoints database [i]

(49, 49+109, 157)-Net over F₃ — Upper bound on s (digital)

There is no digital (49, 158, 158)-net over F₃, because

10 times m-reduction [i] would yield digital (49, 148, 158)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁴⁸, 158, F₃, 99) (dual of [158, 10, 100]-code), but
  - residual code [i] would yield linear OA(3⁴⁹, 58, F₃, 33) (dual of [58, 9, 34]-code), but
    - residual code [i] would yield linear OA(3¹⁶, 24, F₃, 11) (dual of [24, 8, 12]-code), but
      - â€œBSâ€ bound on codes from Brouwerâ€™s database [i]

(49, 49+109, 160)-Net in Base 3 — Upper bound on s

There is no (49, 158, 161)-net in base 3, because

2 times m-reduction [i] would yield (49, 156, 161)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹⁵⁶, 161, S₃, 107), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 809 164816 771822 689786 320611 221860 560835 816670 552324 143733 808294 394923 420563 / 2 > 3¹⁵⁶ [i]