Best Known (52−9, 52, s)-Nets in Base 25
(52−9, 52, 2104966)-Net over F25 — Constructive and digital
Digital (43, 52, 2104966)-net over F25, using
- (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 (32, 41, 2097150)-net over F25, using
- net defined by OOA [i] based on linear OOA(2541, 2097150, F25, 9, 9) (dual of [(2097150, 9), 18874309, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2541, 8388601, F25, 9) (dual of [8388601, 8388560, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(2541, large, F25, 9) (dual of [large, large−41, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2541, large, F25, 9) (dual of [large, large−41, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2541, 8388601, F25, 9) (dual of [8388601, 8388560, 10]-code), using
- net defined by OOA [i] based on linear OOA(2541, 2097150, F25, 9, 9) (dual of [(2097150, 9), 18874309, 10]-NRT-code), using
- digital (7, 11, 7816)-net over F25, using
(52−9, 52, large)-Net over F25 — Digital
Digital (43, 52, large)-net over F25, using
- 251 times duplication [i] based on digital (42, 51, large)-net over F25, using
- t-expansion [i] based on digital (40, 51, large)-net over F25, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2551, large, F25, 11) (dual of [large, large−51, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2551, large, F25, 11) (dual of [large, large−51, 12]-code), using
- t-expansion [i] based on digital (40, 51, large)-net over F25, using
(52−9, 52, large)-Net in Base 25 — Upper bound on s
There is no (43, 52, large)-net in base 25, because
- 7 times m-reduction [i] would yield (43, 45, large)-net in base 25, but