MinT - Best Known (21, 21+5, s)-Nets in Base 3

Best Known (21, 21+5, s)-Nets in Base 3

(21, 21+5, 6480)-Net over F₃ — Constructive and digital

Digital (21, 26, 6480)-net over F₃, using

net defined by OOA [i] based on linear OOA(3²⁶, 6480, F₃, 5, 5) (dual of [(6480, 5), 32374, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(3²⁶, 12961, F₃, 5) (dual of [12961, 12935, 6]-code), using
  - discarding factors / shortening the dual code based on linear OA(3²⁶, 12962, F₃, 5) (dual of [12962, 12936, 6]-code), using
    - trace code [i] based on linear OA(9¹³, 6481, F₉, 5) (dual of [6481, 6468, 6]-code), using
      - base reduction for projective spaces (embedding PG(6,81) in PG(12,9)) [i] based on linear OA(81⁷, 6481, F₈₁, 5) (dual of [6481, 6474, 6]-code), using
        a code by Danev and Olsson [i]

(21, 21+5, 8595)-Net over F₃ — Digital

Digital (21, 26, 8595)-net over F₃, using

net defined by OOA [i] based on linear OOA(3²⁶, 8595, F₃, 5, 5) (dual of [(8595, 5), 42949, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(3²⁶, 8595, F₃, 4, 5) (dual of [(8595, 4), 34354, 6]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁶, 8595, F₃, 5) (dual of [8595, 8569, 6]-code), using
    - discarding factors / shortening the dual code based on linear OA(3²⁶, 12962, F₃, 5) (dual of [12962, 12936, 6]-code), using
      - trace code [i] based on linear OA(9¹³, 6481, F₉, 5) (dual of [6481, 6468, 6]-code), using
        base reduction for projective spaces (embedding PG(6,81) in PG(12,9)) [i] based on linear OA(81⁷, 6481, F₈₁, 5) (dual of [6481, 6474, 6]-code), using
        a code by Danev and Olsson [i]

(21, 21+5, 650878)-Net in Base 3 — Upper bound on s

There is no (21, 26, 650879)-net in base 3, because

1 times m-reduction [i] would yield (21, 25, 650879)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 847290 850557 > 3²⁵ [i]