Best Known (59−10, 59, s)-Nets in Base 25
(59−10, 59, 1685533)-Net over F25 — Constructive and digital
Digital (49, 59, 1685533)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (8, 13, 7813)-net over F25, using
- net defined by OOA [i] based on linear OOA(2513, 7813, F25, 5, 5) (dual of [(7813, 5), 39052, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(2513, 15627, F25, 5) (dual of [15627, 15614, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(2513, 15628, F25, 5) (dual of [15628, 15615, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(2513, 15625, F25, 5) (dual of [15625, 15612, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- 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(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(4) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(2513, 15628, F25, 5) (dual of [15628, 15615, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(2513, 15627, F25, 5) (dual of [15627, 15614, 6]-code), using
- net defined by OOA [i] based on linear OOA(2513, 7813, F25, 5, 5) (dual of [(7813, 5), 39052, 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 (8, 13, 7813)-net over F25, using
(59−10, 59, large)-Net over F25 — Digital
Digital (49, 59, large)-net over F25, using
- t-expansion [i] based on digital (48, 59, large)-net over F25, using
- 2 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
- 2 times m-reduction [i] based on digital (48, 61, large)-net over F25, using
(59−10, 59, large)-Net in Base 25 — Upper bound on s
There is no (49, 59, large)-net in base 25, because
- 8 times m-reduction [i] would yield (49, 51, large)-net in base 25, but