Best Known (56, 85, s)-Nets in Base 25
(56, 85, 1116)-Net over F25 — Constructive and digital
Digital (56, 85, 1116)-net over F25, using
- 253 times duplication [i] based on digital (53, 82, 1116)-net over F25, using
- net defined by OOA [i] based on linear OOA(2582, 1116, F25, 29, 29) (dual of [(1116, 29), 32282, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2582, 15625, F25, 29) (dual of [15625, 15543, 30]-code), using
- an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- OOA 14-folding and stacking with additional row [i] based on linear OA(2582, 15625, F25, 29) (dual of [15625, 15543, 30]-code), using
- net defined by OOA [i] based on linear OOA(2582, 1116, F25, 29, 29) (dual of [(1116, 29), 32282, 30]-NRT-code), using
(56, 85, 10157)-Net over F25 — Digital
Digital (56, 85, 10157)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2585, 10157, F25, 29) (dual of [10157, 10072, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2585, 15626, F25, 29) (dual of [15626, 15541, 30]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2585, 15626, F25, 29) (dual of [15626, 15541, 30]-code), using
(56, 85, large)-Net in Base 25 — Upper bound on s
There is no (56, 85, large)-net in base 25, because
- 27 times m-reduction [i] would yield (56, 58, large)-net in base 25, but