MinT - Best Known (31−23, 31, s)-Nets in Base 3

Best Known (31−23, 31, s)-Nets in Base 3

(31−23, 31, 16)-Net over F₃ — Constructive and digital

Digital (8, 31, 16)-net over F₃, using

t-expansion [i] based on digital (7, 31, 16)-net over F₃, using
- net from sequence [i] based on digital (7, 15)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 7 and N(F) ≥ 16, using
    - an explicitly constructive algebraic function field [i]

(31−23, 31, 17)-Net over F₃ — Digital

Digital (8, 31, 17)-net over F₃, using

net from sequence [i] based on digital (8, 16)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 8 and N(F) ≥ 17, using
  - a curve from the manYPoints database [i]

(31−23, 31, 30)-Net over F₃ — Upper bound on s (digital)

There is no digital (8, 31, 31)-net over F₃, because

5 times m-reduction [i] would yield digital (8, 26, 31)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²⁶, 31, F₃, 18) (dual of [31, 5, 19]-code), but
  - residual code [i] would yield linear OA(3⁸, 12, F₃, 6) (dual of [12, 4, 7]-code), but
    - residual code [i] would yield linear OA(3², 5, F₃, 2) (dual of [5, 3, 3]-code or 5-arc in PG(1,3)), but
      - improvement by Maruta on the Griesmer bound [i]

(31−23, 31, 32)-Net in Base 3 — Upper bound on s

There is no (8, 31, 33)-net in base 3, because

1 times m-reduction [i] would yield (8, 30, 33)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3³⁰, 33, S₃, 22), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 5559 060566 555523 / 23 > 3³⁰ [i]