Best Known (96−17, 96, s)-Nets in Base 25
(96−17, 96, 1048732)-Net over F25 — Constructive and digital
Digital (79, 96, 1048732)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (7, 15, 157)-net over F25, using
- net defined by OOA [i] based on linear OOA(2515, 157, F25, 8, 8) (dual of [(157, 8), 1241, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(2515, 628, F25, 8) (dual of [628, 613, 9]-code), using
- construction XX applied to C1 = C([623,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([623,6]) [i] based on
- linear OA(2513, 624, F25, 7) (dual of [624, 611, 8]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,5}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(2513, 624, F25, 7) (dual of [624, 611, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(2515, 624, F25, 8) (dual of [624, 609, 9]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,6}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(2511, 624, F25, 6) (dual of [624, 613, 7]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(250, 2, F25, 0) (dual of [2, 2, 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
- linear OA(250, 2, F25, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([623,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([623,6]) [i] based on
- OA 4-folding and stacking [i] based on linear OA(2515, 628, F25, 8) (dual of [628, 613, 9]-code), using
- net defined by OOA [i] based on linear OOA(2515, 157, F25, 8, 8) (dual of [(157, 8), 1241, 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 (7, 15, 157)-net over F25, using
(96−17, 96, large)-Net over F25 — Digital
Digital (79, 96, large)-net over F25, using
- t-expansion [i] based on digital (77, 96, large)-net over F25, using
- 1 times m-reduction [i] based on digital (77, 97, large)-net over F25, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2597, large, F25, 20) (dual of [large, large−97, 21]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2596, large, F25, 20) (dual of [large, large−96, 21]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 1 times code embedding in larger space [i] based on linear OA(2596, large, F25, 20) (dual of [large, large−96, 21]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2597, large, F25, 20) (dual of [large, large−97, 21]-code), using
- 1 times m-reduction [i] based on digital (77, 97, large)-net over F25, using
(96−17, 96, large)-Net in Base 25 — Upper bound on s
There is no (79, 96, large)-net in base 25, because
- 15 times m-reduction [i] would yield (79, 81, large)-net in base 25, but