MinT - Best Known (27−5, 27, s)-Nets in Base 3

Best Known (27−5, 27, s)-Nets in Base 3

(27−5, 27, 6481)-Net over F₃ — Constructive and digital

Digital (22, 27, 6481)-net over F₃, using

net defined by OOA [i] based on linear OOA(3²⁷, 6481, F₃, 5, 5) (dual of [(6481, 5), 32378, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(3²⁷, 12963, F₃, 5) (dual of [12963, 12936, 6]-code), using
  - 1 times code embedding in larger space [i] 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]

(27−5, 27, 12398)-Net over F₃ — Digital

Digital (22, 27, 12398)-net over F₃, using

net defined by OOA [i] based on linear OOA(3²⁷, 12398, F₃, 5, 5) (dual of [(12398, 5), 61963, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(3²⁷, 12398, F₃, 4, 5) (dual of [(12398, 4), 49565, 6]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁷, 12398, F₃, 5) (dual of [12398, 12371, 6]-code), using
    - discarding factors / shortening the dual code based on linear OA(3²⁷, 12962, F₃, 5) (dual of [12962, 12935, 6]-code), using
      - base reduction for projective spaces (embedding PG(13,9) in PG(26,3)) [i] based on linear OA(9¹⁴, 12962, F₉, 5) (dual of [12962, 12948, 6]-code), using
        trace code [i] based on linear OA(81⁷, 6481, F₈₁, 5) (dual of [6481, 6474, 6]-code), using
        a code by Danev and Olsson [i]

(27−5, 27, 1127355)-Net in Base 3 — Upper bound on s

There is no (22, 27, 1127356)-net in base 3, because

1 times m-reduction [i] would yield (22, 26, 1127356)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 541869 865609 > 3²⁶ [i]