Best Known (82, 82+17, s)-Nets in Base 25
(82, 82+17, 1048734)-Net over F25 — Constructive and digital
Digital (82, 99, 1048734)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (10, 18, 159)-net over F25, using
- net defined by OOA [i] based on linear OOA(2518, 159, F25, 8, 8) (dual of [(159, 8), 1254, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(2518, 636, F25, 8) (dual of [636, 618, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(3) [i] based on
- linear OA(2515, 625, F25, 8) (dual of [625, 610, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(257, 625, F25, 4) (dual of [625, 618, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(253, 11, F25, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,25) or 11-cap in PG(2,25)), using
- discarding factors / shortening the dual code based on linear OA(253, 25, F25, 3) (dual of [25, 22, 4]-code or 25-arc in PG(2,25) or 25-cap in PG(2,25)), using
- Reed–Solomon code RS(22,25) [i]
- discarding factors / shortening the dual code based on linear OA(253, 25, F25, 3) (dual of [25, 22, 4]-code or 25-arc in PG(2,25) or 25-cap in PG(2,25)), using
- construction X applied to Ce(7) ⊂ Ce(3) [i] based on
- OA 4-folding and stacking [i] based on linear OA(2518, 636, F25, 8) (dual of [636, 618, 9]-code), using
- net defined by OOA [i] based on linear OOA(2518, 159, F25, 8, 8) (dual of [(159, 8), 1254, 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 (10, 18, 159)-net over F25, using
(82, 82+17, large)-Net over F25 — Digital
Digital (82, 99, large)-net over F25, using
- t-expansion [i] based on digital (81, 99, large)-net over F25, using
- 3 times m-reduction [i] based on digital (81, 102, large)-net over F25, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25102, large, F25, 21) (dual of [large, large−102, 22]-code), using
- 1 times code embedding in larger space [i] based on linear OA(25101, large, F25, 21) (dual of [large, large−101, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- 1 times code embedding in larger space [i] based on linear OA(25101, large, F25, 21) (dual of [large, large−101, 22]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25102, large, F25, 21) (dual of [large, large−102, 22]-code), using
- 3 times m-reduction [i] based on digital (81, 102, large)-net over F25, using
(82, 82+17, large)-Net in Base 25 — Upper bound on s
There is no (82, 99, large)-net in base 25, because
- 15 times m-reduction [i] would yield (82, 84, large)-net in base 25, but