Best Known (85, 98, s)-Nets in Base 25
(85, 98, 2804016)-Net over F25 — Constructive and digital
Digital (85, 98, 2804016)-net over F25, using
- generalized (u, u+v)-construction [i] based on
- digital (7, 11, 7816)-net over F25, using
- net defined by OOA [i] based on linear OOA(2511, 7816, F25, 4, 4) (dual of [(7816, 4), 31253, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(2511, 15632, F25, 4) (dual of [15632, 15621, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(1) [i] based on
- linear OA(2510, 15625, F25, 4) (dual of [15625, 15615, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(254, 15625, F25, 2) (dual of [15625, 15621, 3]-code), using an extension Ce(1) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,1], and designed minimum distance d ≥ |I|+1 = 2 [i]
- linear OA(251, 7, F25, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(251, s, F25, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(1) [i] based on
- OA 2-folding and stacking [i] based on linear OA(2511, 15632, F25, 4) (dual of [15632, 15621, 5]-code), using
- net defined by OOA [i] based on linear OOA(2511, 7816, F25, 4, 4) (dual of [(7816, 4), 31253, 5]-NRT-code), using
- digital (20, 26, 1398100)-net over F25, using
- s-reduction based on digital (20, 26, 2796201)-net over F25, using
- net defined by OOA [i] based on linear OOA(2526, 2796201, F25, 6, 6) (dual of [(2796201, 6), 16777180, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(2526, large, F25, 6) (dual of [large, large−26, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OA 3-folding and stacking [i] based on linear OA(2526, large, F25, 6) (dual of [large, large−26, 7]-code), using
- net defined by OOA [i] based on linear OOA(2526, 2796201, F25, 6, 6) (dual of [(2796201, 6), 16777180, 7]-NRT-code), using
- s-reduction based on digital (20, 26, 2796201)-net over F25, using
- digital (48, 61, 1398100)-net over F25, using
- net defined by OOA [i] based on linear OOA(2561, 1398100, F25, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2561, 8388601, F25, 13) (dual of [8388601, 8388540, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, large, F25, 13) (dual of [large, large−61, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2561, large, F25, 13) (dual of [large, large−61, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2561, 8388601, F25, 13) (dual of [8388601, 8388540, 14]-code), using
- net defined by OOA [i] based on linear OOA(2561, 1398100, F25, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
- digital (7, 11, 7816)-net over F25, using
(85, 98, large)-Net over F25 — Digital
Digital (85, 98, large)-net over F25, using
- 9 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
(85, 98, large)-Net in Base 25 — Upper bound on s
There is no (85, 98, large)-net in base 25, because
- 11 times m-reduction [i] would yield (85, 87, large)-net in base 25, but