Best Known (50, 60, s)-Nets in Base 25
(50, 60, 1685536)-Net over F25 — Constructive and digital
Digital (50, 60, 1685536)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (9, 14, 7816)-net over F25, using
- net defined by OOA [i] based on linear OOA(2514, 7816, F25, 5, 5) (dual of [(7816, 5), 39066, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(2514, 15633, F25, 5) (dual of [15633, 15619, 6]-code), using
- construction X applied to C([0,2]) ⊂ C([0,1]) [i] based on
- linear OA(2513, 15626, F25, 5) (dual of [15626, 15613, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(257, 15626, F25, 3) (dual of [15626, 15619, 4]-code or 15626-cap in PG(6,25)), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,1], and minimum distance d ≥ |{−1,0,1}|+1 = 4 (BCH-bound) [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 C([0,2]) ⊂ C([0,1]) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(2514, 15633, F25, 5) (dual of [15633, 15619, 6]-code), using
- net defined by OOA [i] based on linear OOA(2514, 7816, F25, 5, 5) (dual of [(7816, 5), 39066, 6]-NRT-code), using
- 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 (9, 14, 7816)-net over F25, using
(50, 60, large)-Net over F25 — Digital
Digital (50, 60, large)-net over F25, using
- t-expansion [i] based on digital (48, 60, large)-net over F25, using
- 1 times m-reduction [i] based on digital (48, 61, large)-net over F25, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] 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]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2561, large, F25, 13) (dual of [large, large−61, 14]-code), using
- 1 times m-reduction [i] based on digital (48, 61, large)-net over F25, using
(50, 60, large)-Net in Base 25 — Upper bound on s
There is no (50, 60, large)-net in base 25, because
- 8 times m-reduction [i] would yield (50, 52, large)-net in base 25, but