Best Known (61, 61+10, s)-Nets in Base 25
(61, 61+10, 3356066)-Net over F25 — Constructive and digital
Digital (61, 71, 3356066)-net over F25, using
- generalized (u, u+v)-construction [i] based on
- digital (1, 4, 626)-net over F25, using
- net defined by OOA [i] based on linear OOA(254, 626, F25, 3, 3) (dual of [(626, 3), 1874, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(254, 626, F25, 2, 3) (dual of [(626, 2), 1248, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(254, 626, F25, 3, 3) (dual of [(626, 3), 1874, 4]-NRT-code), using
- digital (16, 21, 1677720)-net over F25, using
- s-reduction based on digital (16, 21, 4194301)-net over F25, using
- net defined by OOA [i] based on linear OOA(2521, 4194301, F25, 5, 5) (dual of [(4194301, 5), 20971484, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(2521, large, F25, 5) (dual of [large, large−21, 6]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- OOA 2-folding and stacking with additional row [i] based on linear OA(2521, large, F25, 5) (dual of [large, large−21, 6]-code), using
- net defined by OOA [i] based on linear OOA(2521, 4194301, F25, 5, 5) (dual of [(4194301, 5), 20971484, 6]-NRT-code), using
- s-reduction based on digital (16, 21, 4194301)-net over F25, 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 (1, 4, 626)-net over F25, using
(61, 61+10, large)-Net over F25 — Digital
Digital (61, 71, large)-net over F25, using
- 6 times m-reduction [i] based on digital (61, 77, large)-net over F25, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2577, large, F25, 16) (dual of [large, large−77, 17]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2576, large, F25, 16) (dual of [large, large−76, 17]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 1 times code embedding in larger space [i] based on linear OA(2576, large, F25, 16) (dual of [large, large−76, 17]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2577, large, F25, 16) (dual of [large, large−77, 17]-code), using
(61, 61+10, large)-Net in Base 25 — Upper bound on s
There is no (61, 71, large)-net in base 25, because
- 8 times m-reduction [i] would yield (61, 63, large)-net in base 25, but