Best Known (87, 87+15, s)-Nets in Base 25
(87, 87+15, 2396742)-Net over F25 — Constructive and digital
Digital (87, 102, 2396742)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (24, 31, 2796200)-net over F25, using
- net defined by OOA [i] based on linear OOA(2531, 2796200, F25, 7, 7) (dual of [(2796200, 7), 19573369, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2531, 8388601, F25, 7) (dual of [8388601, 8388570, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(2531, large, F25, 7) (dual of [large, large−31, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2531, large, F25, 7) (dual of [large, large−31, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2531, 8388601, F25, 7) (dual of [8388601, 8388570, 8]-code), using
- net defined by OOA [i] based on linear OOA(2531, 2796200, F25, 7, 7) (dual of [(2796200, 7), 19573369, 8]-NRT-code), using
- digital (56, 71, 1198371)-net over F25, using
- net defined by OOA [i] based on linear OOA(2571, 1198371, F25, 15, 15) (dual of [(1198371, 15), 17975494, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2571, 8388598, F25, 15) (dual of [8388598, 8388527, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2571, large, F25, 15) (dual of [large, large−71, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2571, large, F25, 15) (dual of [large, large−71, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2571, 8388598, F25, 15) (dual of [8388598, 8388527, 16]-code), using
- net defined by OOA [i] based on linear OOA(2571, 1198371, F25, 15, 15) (dual of [(1198371, 15), 17975494, 16]-NRT-code), using
- digital (24, 31, 2796200)-net over F25, using
(87, 87+15, large)-Net over F25 — Digital
Digital (87, 102, large)-net over F25, using
- t-expansion [i] based on digital (85, 102, large)-net over F25, using
- 5 times m-reduction [i] based on digital (85, 107, large)-net over F25, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25107, large, F25, 22) (dual of [large, large−107, 23]-code), using
- 1 times code embedding in larger space [i] based on linear OA(25106, large, F25, 22) (dual of [large, large−106, 23]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- 1 times code embedding in larger space [i] based on linear OA(25106, large, F25, 22) (dual of [large, large−106, 23]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25107, large, F25, 22) (dual of [large, large−107, 23]-code), using
- 5 times m-reduction [i] based on digital (85, 107, large)-net over F25, using
(87, 87+15, large)-Net in Base 25 — Upper bound on s
There is no (87, 102, large)-net in base 25, because
- 13 times m-reduction [i] would yield (87, 89, large)-net in base 25, but