Best Known (25−5, 25, s)-Nets in Base 3
(25−5, 25, 3284)-Net over F3 — Constructive and digital
Digital (20, 25, 3284)-net over F3, using
- net defined by OOA [i] based on linear OOA(325, 3284, F3, 5, 5) (dual of [(3284, 5), 16395, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(325, 6569, F3, 5) (dual of [6569, 6544, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(325, 6561, F3, 5) (dual of [6561, 6536, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(317, 6561, F3, 4) (dual of [6561, 6544, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(30, 8, F3, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(325, 6569, F3, 5) (dual of [6569, 6544, 6]-code), using
(25−5, 25, 5959)-Net over F3 — Digital
Digital (20, 25, 5959)-net over F3, using
- net defined by OOA [i] based on linear OOA(325, 5959, F3, 5, 5) (dual of [(5959, 5), 29770, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(325, 5959, F3, 4, 5) (dual of [(5959, 4), 23811, 6]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(325, 5959, F3, 5) (dual of [5959, 5934, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(325, 6561, F3, 5) (dual of [6561, 6536, 6]-code), using
- an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(325, 6561, F3, 5) (dual of [6561, 6536, 6]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(325, 5959, F3, 5) (dual of [5959, 5934, 6]-code), using
- appending kth column [i] based on linear OOA(325, 5959, F3, 4, 5) (dual of [(5959, 4), 23811, 6]-NRT-code), using
(25−5, 25, 375784)-Net in Base 3 — Upper bound on s
There is no (20, 25, 375785)-net in base 3, because
- 1 times m-reduction [i] would yield (20, 24, 375785)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 282430 987161 > 324 [i]