Best Known (92−12, 92, s)-Nets in Base 25
(92−12, 92, 2804014)-Net over F25 — Constructive and digital
Digital (80, 92, 2804014)-net over F25, using
- generalized (u, u+v)-construction [i] based on
- digital (6, 10, 7814)-net over F25, using
- net defined by OOA [i] based on linear OOA(2510, 7814, F25, 4, 4) (dual of [(7814, 4), 31246, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(2510, 15628, F25, 4) (dual of [15628, 15618, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [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(257, 15625, F25, 3) (dual of [15625, 15618, 4]-code or 15625-cap in PG(6,25)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [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(3) ⊂ Ce(2) [i] based on
- OA 2-folding and stacking [i] based on linear OA(2510, 15628, F25, 4) (dual of [15628, 15618, 5]-code), using
- net defined by OOA [i] based on linear OOA(2510, 7814, F25, 4, 4) (dual of [(7814, 4), 31246, 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 (44, 56, 1398100)-net over F25, using
- net defined by OOA [i] based on linear OOA(2556, 1398100, F25, 12, 12) (dual of [(1398100, 12), 16777144, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(2556, 8388600, F25, 12) (dual of [8388600, 8388544, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(2556, large, F25, 12) (dual of [large, large−56, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(2556, large, F25, 12) (dual of [large, large−56, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(2556, 8388600, F25, 12) (dual of [8388600, 8388544, 13]-code), using
- net defined by OOA [i] based on linear OOA(2556, 1398100, F25, 12, 12) (dual of [(1398100, 12), 16777144, 13]-NRT-code), using
- digital (6, 10, 7814)-net over F25, using
(92−12, 92, large)-Net over F25 — Digital
Digital (80, 92, large)-net over F25, using
- t-expansion [i] based on digital (77, 92, large)-net over F25, using
- 5 times m-reduction [i] based on digital (77, 97, large)-net over F25, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2597, large, F25, 20) (dual of [large, large−97, 21]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2596, large, F25, 20) (dual of [large, large−96, 21]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 1 times code embedding in larger space [i] based on linear OA(2596, large, F25, 20) (dual of [large, large−96, 21]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2597, large, F25, 20) (dual of [large, large−97, 21]-code), using
- 5 times m-reduction [i] based on digital (77, 97, large)-net over F25, using
(92−12, 92, large)-Net in Base 25 — Upper bound on s
There is no (80, 92, large)-net in base 25, because
- 10 times m-reduction [i] would yield (80, 82, large)-net in base 25, but