Best Known (51−13, 51, s)-Nets in Base 25
(51−13, 51, 65106)-Net over F25 — Constructive and digital
Digital (38, 51, 65106)-net over F25, using
- net defined by OOA [i] based on linear OOA(2551, 65106, F25, 13, 13) (dual of [(65106, 13), 846327, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2551, 390637, F25, 13) (dual of [390637, 390586, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(2551, 390639, F25, 13) (dual of [390639, 390588, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(2549, 390625, F25, 13) (dual of [390625, 390576, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2537, 390625, F25, 10) (dual of [390625, 390588, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(12) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(2551, 390639, F25, 13) (dual of [390639, 390588, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2551, 390637, F25, 13) (dual of [390637, 390586, 14]-code), using
(51−13, 51, 390639)-Net over F25 — Digital
Digital (38, 51, 390639)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2551, 390639, F25, 13) (dual of [390639, 390588, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(2549, 390625, F25, 13) (dual of [390625, 390576, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2537, 390625, F25, 10) (dual of [390625, 390588, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(12) ⊂ Ce(9) [i] based on
(51−13, 51, large)-Net in Base 25 — Upper bound on s
There is no (38, 51, large)-net in base 25, because
- 11 times m-reduction [i] would yield (38, 40, large)-net in base 25, but