Best Known (54, 54+18, s)-Nets in Base 25
(54, 54+18, 43404)-Net over F25 — Constructive and digital
Digital (54, 72, 43404)-net over F25, using
- 251 times duplication [i] based on digital (53, 71, 43404)-net over F25, using
- net defined by OOA [i] based on linear OOA(2571, 43404, F25, 18, 18) (dual of [(43404, 18), 781201, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(2571, 390636, F25, 18) (dual of [390636, 390565, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2571, 390639, F25, 18) (dual of [390639, 390568, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- linear OA(2569, 390625, F25, 18) (dual of [390625, 390556, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(2557, 390625, F25, 15) (dual of [390625, 390568, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(17) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(2571, 390639, F25, 18) (dual of [390639, 390568, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(2571, 390636, F25, 18) (dual of [390636, 390565, 19]-code), using
- net defined by OOA [i] based on linear OOA(2571, 43404, F25, 18, 18) (dual of [(43404, 18), 781201, 19]-NRT-code), using
(54, 54+18, 390644)-Net over F25 — Digital
Digital (54, 72, 390644)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2572, 390644, F25, 18) (dual of [390644, 390572, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- linear OA(2569, 390625, F25, 18) (dual of [390625, 390556, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(2553, 390625, F25, 14) (dual of [390625, 390572, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(253, 19, F25, 3) (dual of [19, 16, 4]-code or 19-arc in PG(2,25) or 19-cap in PG(2,25)), using
- discarding factors / shortening the dual code based on linear OA(253, 25, F25, 3) (dual of [25, 22, 4]-code or 25-arc in PG(2,25) or 25-cap in PG(2,25)), using
- Reed–Solomon code RS(22,25) [i]
- discarding factors / shortening the dual code based on linear OA(253, 25, F25, 3) (dual of [25, 22, 4]-code or 25-arc in PG(2,25) or 25-cap in PG(2,25)), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
(54, 54+18, large)-Net in Base 25 — Upper bound on s
There is no (54, 72, large)-net in base 25, because
- 16 times m-reduction [i] would yield (54, 56, large)-net in base 25, but