Best Known (19−7, 19, s)-Nets in Base 25
(19−7, 19, 5209)-Net over F25 — Constructive and digital
Digital (12, 19, 5209)-net over F25, using
- net defined by OOA [i] based on linear OOA(2519, 5209, F25, 7, 7) (dual of [(5209, 7), 36444, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2519, 15628, F25, 7) (dual of [15628, 15609, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(2519, 15625, F25, 7) (dual of [15625, 15606, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(2516, 15625, F25, 6) (dual of [15625, 15609, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(250, 3, F25, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(250, s, F25, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(2519, 15628, F25, 7) (dual of [15628, 15609, 8]-code), using
(19−7, 19, 11699)-Net over F25 — Digital
Digital (12, 19, 11699)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2519, 11699, F25, 7) (dual of [11699, 11680, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(2519, 15625, F25, 7) (dual of [15625, 15606, 8]-code), using
- an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(2519, 15625, F25, 7) (dual of [15625, 15606, 8]-code), using
(19−7, 19, large)-Net in Base 25 — Upper bound on s
There is no (12, 19, large)-net in base 25, because
- 5 times m-reduction [i] would yield (12, 14, large)-net in base 25, but