MinT - Best Known (72−57, 72, s)-Nets in Base 4

Best Known (72−57, 72, s)-Nets in Base 4

(72−57, 72, 33)-Net over F₄ — Constructive and digital

Digital (15, 72, 33)-net over F₄, using

net from sequence [i] based on digital (15, 32)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
  - an explicitly constructive algebraic function field [i]

(72−57, 72, 35)-Net over F₄ — Digital

Digital (15, 72, 35)-net over F₄, using

net from sequence [i] based on digital (15, 34)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 35, using
  - a curve from the manYPoints database [i]

(72−57, 72, 67)-Net over F₄ — Upper bound on s (digital)

There is no digital (15, 72, 68)-net over F₄, because

9 times m-reduction [i] would yield digital (15, 63, 68)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4⁶³, 68, F₄, 48) (dual of [68, 5, 49]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(4⁶², 65, F₄, 48) (dual of [65, 3, 49]-code), but
      - Griesmer bound [i]
    2. linear OA(4⁵, 68, F₄, 3) (dual of [68, 63, 4]-code or 68-cap in PG(4,4)), but
      - discarding factors / shortening the dual code would yield linear OA(4⁵, 42, F₄, 3) (dual of [42, 37, 4]-code or 42-cap in PG(4,4)), but
        41 is the largest size of a cap in PG(4,Â 4) [i]

(72−57, 72, 69)-Net in Base 4 — Upper bound on s

There is no (15, 72, 70)-net in base 4, because

9 times m-reduction [i] would yield (15, 63, 70)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4⁶³, 70, S₄, 48), but
  - the linear programming bound shows that MÂ â‰¥ 87112 285931 760246 646623 899502 532662 132736 / 931 > 4⁶³ [i]