MinT - Best Known (23, 28, s)-Nets in Base 3

Best Known (23, 28, s)-Nets in Base 3

Digital (23, 28, 12961)-net over F₃, using

net defined by OOA [i] based on linear OOA(3²⁸, 12961, F₃, 5, 5) (dual of [(12961, 5), 64777, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(3²⁸, 25923, F₃, 5) (dual of [25923, 25895, 6]-code), using
  - discarding factors / shortening the dual code based on linear OA(3²⁸, 25924, F₃, 5) (dual of [25924, 25896, 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]

Digital (23, 28, 17881)-net over F₃, using

net defined by OOA [i] based on linear OOA(3²⁸, 17881, F₃, 5, 5) (dual of [(17881, 5), 89377, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(3²⁸, 17881, F₃, 4, 5) (dual of [(17881, 4), 71496, 6]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁸, 17881, F₃, 5) (dual of [17881, 17853, 6]-code), using
    - discarding factors / shortening the dual code based on linear OA(3²⁸, 19683, F₃, 5) (dual of [19683, 19655, 6]-code), using
      - an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]

There is no (23, 28, 1952638)-net in base 3, because

1 times m-reduction [i] would yield (23, 27, 1952638)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 7 625602 033917 > 3²⁷ [i]