Best Known (55−18, 55, s)-Nets in Base 25
(55−18, 55, 1737)-Net over F25 — Constructive and digital
Digital (37, 55, 1737)-net over F25, using
- 251 times duplication [i] based on digital (36, 54, 1737)-net over F25, using
- net defined by OOA [i] based on linear OOA(2554, 1737, F25, 18, 18) (dual of [(1737, 18), 31212, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(2554, 15633, F25, 18) (dual of [15633, 15579, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2554, 15636, F25, 18) (dual of [15636, 15582, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- linear OA(2552, 15625, F25, 18) (dual of [15625, 15573, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(2543, 15625, F25, 15) (dual of [15625, 15582, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(252, 11, F25, 2) (dual of [11, 9, 3]-code or 11-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(2554, 15636, F25, 18) (dual of [15636, 15582, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(2554, 15633, F25, 18) (dual of [15633, 15579, 19]-code), using
- net defined by OOA [i] based on linear OOA(2554, 1737, F25, 18, 18) (dual of [(1737, 18), 31212, 19]-NRT-code), using
(55−18, 55, 14796)-Net over F25 — Digital
Digital (37, 55, 14796)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2555, 14796, F25, 18) (dual of [14796, 14741, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2555, 15640, F25, 18) (dual of [15640, 15585, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- linear OA(2552, 15625, F25, 18) (dual of [15625, 15573, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(2540, 15625, F25, 14) (dual of [15625, 15585, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(253, 15, F25, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,25) or 15-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
- discarding factors / shortening the dual code based on linear OA(2555, 15640, F25, 18) (dual of [15640, 15585, 19]-code), using
(55−18, 55, large)-Net in Base 25 — Upper bound on s
There is no (37, 55, large)-net in base 25, because
- 16 times m-reduction [i] would yield (37, 39, large)-net in base 25, but