Best Known (57, 68, s)-Nets in Base 25
(57, 68, 1873034)-Net over F25 — Constructive and digital
Digital (57, 68, 1873034)-net over F25, using
- (u, u+v)-construction [i] based on
- 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
- digital (40, 51, 1677720)-net over F25, using
- net defined by OOA [i] based on linear OOA(2551, 1677720, F25, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2551, 8388601, F25, 11) (dual of [8388601, 8388550, 12]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(2551, large, F25, 11) (dual of [large, large−51, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2551, 8388601, F25, 11) (dual of [8388601, 8388550, 12]-code), using
- net defined by OOA [i] based on linear OOA(2551, 1677720, F25, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- digital (12, 17, 195314)-net over F25, using
(57, 68, large)-Net over F25 — Digital
Digital (57, 68, large)-net over F25, using
- 4 times m-reduction [i] based on digital (57, 72, large)-net over F25, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2572, large, F25, 15) (dual of [large, large−72, 16]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2571, large, F25, 15) (dual of [large, large−71, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- 1 times code embedding in larger space [i] based on linear OA(2571, large, F25, 15) (dual of [large, large−71, 16]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2572, large, F25, 15) (dual of [large, large−72, 16]-code), using
(57, 68, large)-Net in Base 25 — Upper bound on s
There is no (57, 68, large)-net in base 25, because
- 9 times m-reduction [i] would yield (57, 59, large)-net in base 25, but