Best Known (56, 66, s)-Nets in Base 25
(56, 66, 1873685)-Net over F25 — Constructive and digital
Digital (56, 66, 1873685)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (15, 20, 195965)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (1, 3, 651)-net over F25, using
- digital (12, 17, 195314)-net over F25, using
- net defined by OOA [i] based on linear OOA(2517, 195314, F25, 5, 5) (dual of [(195314, 5), 976553, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(2517, 390629, F25, 5) (dual of [390629, 390612, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(2517, 390625, F25, 5) (dual of [390625, 390608, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(2513, 390625, F25, 4) (dual of [390625, 390612, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(250, 4, F25, 0) (dual of [4, 4, 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(4) ⊂ Ce(3) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(2517, 390629, F25, 5) (dual of [390629, 390612, 6]-code), using
- net defined by OOA [i] based on linear OOA(2517, 195314, F25, 5, 5) (dual of [(195314, 5), 976553, 6]-NRT-code), using
- (u, u+v)-construction [i] based on
- digital (36, 46, 1677720)-net over F25, using
- net defined by OOA [i] based on linear OOA(2546, 1677720, F25, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(2546, 8388600, F25, 10) (dual of [8388600, 8388554, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(2546, large, F25, 10) (dual of [large, large−46, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(2546, large, F25, 10) (dual of [large, large−46, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(2546, 8388600, F25, 10) (dual of [8388600, 8388554, 11]-code), using
- net defined by OOA [i] based on linear OOA(2546, 1677720, F25, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
- digital (15, 20, 195965)-net over F25, using
(56, 66, large)-Net over F25 — Digital
Digital (56, 66, large)-net over F25, using
- t-expansion [i] based on digital (53, 66, large)-net over F25, using
- 1 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
- 1 times m-reduction [i] based on digital (53, 67, large)-net over F25, using
(56, 66, large)-Net in Base 25 — Upper bound on s
There is no (56, 66, large)-net in base 25, because
- 8 times m-reduction [i] would yield (56, 58, large)-net in base 25, but