Best Known (41, 41+15, s)-Nets in Base 25
(41, 41+15, 2441)-Net over F25 — Constructive and digital
Digital (41, 56, 2441)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (6, 13, 209)-net over F25, using
- net defined by OOA [i] based on linear OOA(2513, 209, F25, 7, 7) (dual of [(209, 7), 1450, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2513, 628, F25, 7) (dual of [628, 615, 8]-code), using
- construction XX applied to C1 = C([623,4]), C2 = C([0,5]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([623,5]) [i] based on
- linear OA(2511, 624, F25, 6) (dual of [624, 613, 7]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,4}, and designed minimum distance d ≥ |I|+1 = 7 [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(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(259, 624, F25, 5) (dual of [624, 615, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [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([623,4]), C2 = C([0,5]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([623,5]) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(2513, 628, F25, 7) (dual of [628, 615, 8]-code), using
- net defined by OOA [i] based on linear OOA(2513, 209, F25, 7, 7) (dual of [(209, 7), 1450, 8]-NRT-code), using
- digital (28, 43, 2232)-net over F25, using
- net defined by OOA [i] based on linear OOA(2543, 2232, F25, 15, 15) (dual of [(2232, 15), 33437, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2543, 15625, F25, 15) (dual of [15625, 15582, 16]-code), using
- an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- OOA 7-folding and stacking with additional row [i] based on linear OA(2543, 15625, F25, 15) (dual of [15625, 15582, 16]-code), using
- net defined by OOA [i] based on linear OOA(2543, 2232, F25, 15, 15) (dual of [(2232, 15), 33437, 16]-NRT-code), using
- digital (6, 13, 209)-net over F25, using
(41, 41+15, 98410)-Net over F25 — Digital
Digital (41, 56, 98410)-net over F25, using
(41, 41+15, large)-Net in Base 25 — Upper bound on s
There is no (41, 56, large)-net in base 25, because
- 13 times m-reduction [i] would yield (41, 43, large)-net in base 25, but