Best Known (86, 86+17, s)-Nets in Base 25
(86, 86+17, 1052482)-Net over F25 — Constructive and digital
Digital (86, 103, 1052482)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (14, 22, 3907)-net over F25, using
- net defined by OOA [i] based on linear OOA(2522, 3907, F25, 8, 8) (dual of [(3907, 8), 31234, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(2522, 15628, F25, 8) (dual of [15628, 15606, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(2522, 15625, F25, 8) (dual of [15625, 15603, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(2519, 15625, F25, 7) (dual of [15625, 15606, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(7) ⊂ Ce(6) [i] based on
- OA 4-folding and stacking [i] based on linear OA(2522, 15628, F25, 8) (dual of [15628, 15606, 9]-code), using
- net defined by OOA [i] based on linear OOA(2522, 3907, F25, 8, 8) (dual of [(3907, 8), 31234, 9]-NRT-code), using
- digital (64, 81, 1048575)-net over F25, using
- net defined by OOA [i] based on linear OOA(2581, 1048575, F25, 17, 17) (dual of [(1048575, 17), 17825694, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2581, 8388601, F25, 17) (dual of [8388601, 8388520, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2581, large, F25, 17) (dual of [large, large−81, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2581, large, F25, 17) (dual of [large, large−81, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2581, 8388601, F25, 17) (dual of [8388601, 8388520, 18]-code), using
- net defined by OOA [i] based on linear OOA(2581, 1048575, F25, 17, 17) (dual of [(1048575, 17), 17825694, 18]-NRT-code), using
- digital (14, 22, 3907)-net over F25, using
(86, 86+17, large)-Net over F25 — Digital
Digital (86, 103, large)-net over F25, using
- t-expansion [i] based on digital (85, 103, large)-net over F25, using
- 4 times m-reduction [i] based on digital (85, 107, large)-net over F25, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25107, large, F25, 22) (dual of [large, large−107, 23]-code), using
- 1 times code embedding in larger space [i] based on linear OA(25106, large, F25, 22) (dual of [large, large−106, 23]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- 1 times code embedding in larger space [i] based on linear OA(25106, large, F25, 22) (dual of [large, large−106, 23]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25107, large, F25, 22) (dual of [large, large−107, 23]-code), using
- 4 times m-reduction [i] based on digital (85, 107, large)-net over F25, using
(86, 86+17, large)-Net in Base 25 — Upper bound on s
There is no (86, 103, large)-net in base 25, because
- 15 times m-reduction [i] would yield (86, 88, large)-net in base 25, but