Best Known (56, 56+8, s)-Nets in Base 25
(56, 56+8, 8388600)-Net over F25 — Constructive and digital
Digital (56, 64, 8388600)-net over F25, using
- generalized (u, u+v)-construction [i] based on
- digital (4, 6, 2097150)-net over F25, using
- s-reduction based on digital (4, 6, large)-net over F25, using
- digital (4, 6, 2097150)-net over F25 (see above)
- digital (12, 16, 2097150)-net over F25, using
- s-reduction based on digital (12, 16, 4194301)-net over F25, using
- net defined by OOA [i] based on linear OOA(2516, 4194301, F25, 4, 4) (dual of [(4194301, 4), 16777188, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(2516, 8388602, F25, 4) (dual of [8388602, 8388586, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(2516, large, F25, 4) (dual of [large, large−16, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(2516, large, F25, 4) (dual of [large, large−16, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(2516, 8388602, F25, 4) (dual of [8388602, 8388586, 5]-code), using
- net defined by OOA [i] based on linear OOA(2516, 4194301, F25, 4, 4) (dual of [(4194301, 4), 16777188, 5]-NRT-code), using
- s-reduction based on digital (12, 16, 4194301)-net over F25, using
- digital (28, 36, 2097150)-net over F25, using
- net defined by OOA [i] based on linear OOA(2536, 2097150, F25, 8, 8) (dual of [(2097150, 8), 16777164, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(2536, 8388600, F25, 8) (dual of [8388600, 8388564, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(2536, large, F25, 8) (dual of [large, large−36, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(2536, large, F25, 8) (dual of [large, large−36, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(2536, 8388600, F25, 8) (dual of [8388600, 8388564, 9]-code), using
- net defined by OOA [i] based on linear OOA(2536, 2097150, F25, 8, 8) (dual of [(2097150, 8), 16777164, 9]-NRT-code), using
- digital (4, 6, 2097150)-net over F25, using
(56, 56+8, large)-Net over F25 — Digital
Digital (56, 64, large)-net over F25, using
- t-expansion [i] based on digital (53, 64, large)-net over F25, using
- 3 times m-reduction [i] based on digital (53, 67, large)-net over F25, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2567, large, F25, 14) (dual of [large, large−67, 15]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2566, large, F25, 14) (dual of [large, large−66, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- 1 times code embedding in larger space [i] based on linear OA(2566, large, F25, 14) (dual of [large, large−66, 15]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2567, large, F25, 14) (dual of [large, large−67, 15]-code), using
- 3 times m-reduction [i] based on digital (53, 67, large)-net over F25, using
(56, 56+8, large)-Net in Base 25 — Upper bound on s
There is no (56, 64, large)-net in base 25, because
- 6 times m-reduction [i] would yield (56, 58, large)-net in base 25, but