Best Known (57, 57+19, s)-Nets in Base 25
(57, 57+19, 43404)-Net over F25 — Constructive and digital
Digital (57, 76, 43404)-net over F25, using
- 251 times duplication [i] based on digital (56, 75, 43404)-net over F25, using
- net defined by OOA [i] based on linear OOA(2575, 43404, F25, 19, 19) (dual of [(43404, 19), 824601, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2575, 390637, F25, 19) (dual of [390637, 390562, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2575, 390639, F25, 19) (dual of [390639, 390564, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(2573, 390625, F25, 19) (dual of [390625, 390552, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2561, 390625, F25, 16) (dual of [390625, 390564, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(18) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(2575, 390639, F25, 19) (dual of [390639, 390564, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2575, 390637, F25, 19) (dual of [390637, 390562, 20]-code), using
- net defined by OOA [i] based on linear OOA(2575, 43404, F25, 19, 19) (dual of [(43404, 19), 824601, 20]-NRT-code), using
(57, 57+19, 390645)-Net over F25 — Digital
Digital (57, 76, 390645)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2576, 390645, F25, 19) (dual of [390645, 390569, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- linear OA(2573, 390626, F25, 19) (dual of [390626, 390553, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 390626 | 258−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(2557, 390626, F25, 15) (dual of [390626, 390569, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 390626 | 258−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [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 C([0,9]) ⊂ C([0,7]) [i] based on
(57, 57+19, large)-Net in Base 25 — Upper bound on s
There is no (57, 76, large)-net in base 25, because
- 17 times m-reduction [i] would yield (57, 59, large)-net in base 25, but