Best Known (20, 20+7, s)-Nets in Base 25
(20, 20+7, 130212)-Net over F25 — Constructive and digital
Digital (20, 27, 130212)-net over F25, using
- net defined by OOA [i] based on linear OOA(2527, 130212, F25, 7, 7) (dual of [(130212, 7), 911457, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2527, 390637, F25, 7) (dual of [390637, 390610, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(2527, 390639, F25, 7) (dual of [390639, 390612, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
- linear OA(2525, 390625, F25, 7) (dual of [390625, 390600, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(2513, 390625, F25, 4) (dual of [390625, 390612, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(252, 14, F25, 2) (dual of [14, 12, 3]-code or 14-arc in PG(1,25)), using
- discarding factors / shortening the dual code based on linear OA(252, 25, F25, 2) (dual of [25, 23, 3]-code or 25-arc in PG(1,25)), using
- Reed–Solomon code RS(23,25) [i]
- discarding factors / shortening the dual code based on linear OA(252, 25, F25, 2) (dual of [25, 23, 3]-code or 25-arc in PG(1,25)), using
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(2527, 390639, F25, 7) (dual of [390639, 390612, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2527, 390637, F25, 7) (dual of [390637, 390610, 8]-code), using
(20, 20+7, 390639)-Net over F25 — Digital
Digital (20, 27, 390639)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2527, 390639, F25, 7) (dual of [390639, 390612, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
- linear OA(2525, 390625, F25, 7) (dual of [390625, 390600, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(2513, 390625, F25, 4) (dual of [390625, 390612, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(252, 14, F25, 2) (dual of [14, 12, 3]-code or 14-arc in PG(1,25)), using
- discarding factors / shortening the dual code based on linear OA(252, 25, F25, 2) (dual of [25, 23, 3]-code or 25-arc in PG(1,25)), using
- Reed–Solomon code RS(23,25) [i]
- discarding factors / shortening the dual code based on linear OA(252, 25, F25, 2) (dual of [25, 23, 3]-code or 25-arc in PG(1,25)), using
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
(20, 20+7, large)-Net in Base 25 — Upper bound on s
There is no (20, 27, large)-net in base 25, because
- 5 times m-reduction [i] would yield (20, 22, large)-net in base 25, but