MinT - Best Known (21−15, 21, s)-Nets in Base 3

Best Known (21−15, 21, s)-Nets in Base 3

(21−15, 21, 14)-Net over F₃ — Constructive and digital

Digital (6, 21, 14)-net over F₃, using

net from sequence [i] based on digital (6, 13)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 6 and N(F) ≥ 14, using
  - an explicitly constructive algebraic function field [i]

(21−15, 21, 25)-Net over F₃ — Upper bound on s (digital)

There is no digital (6, 21, 26)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²¹, 26, F₃, 15) (dual of [26, 5, 16]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(3²⁰, 23, S₃, 15), but
    - the (dual) Plotkin bound shows that MÂ â‰¥ 31381 059609 / 8 > 3²⁰ [i]
  2. linear OA(3⁵, 26, F₃, 3) (dual of [26, 21, 4]-code or 26-cap in PG(4,3)), but
    - discarding factors / shortening the dual code would yield linear OA(3⁵, 21, F₃, 3) (dual of [21, 16, 4]-code or 21-cap in PG(4,3)), but
      - 20 is the largest size of a cap in PG(4,Â 3) [i]

(21−15, 21, 26)-Net in Base 3 — Upper bound on s

There is no (6, 21, 27)-net in base 3, because

extracting embedded OOA [i] would yield OOA(3²¹, 27, S₃, 2, 15), but
- the linear programming bound for OOAs shows that MÂ â‰¥ 503297 059606 626987 / 44 441752 > 3²¹ [i]