Best Known (85−15, 85, s)-Nets in Base 25
(85−15, 85, 1198581)-Net over F25 — Constructive and digital
Digital (70, 85, 1198581)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (7, 14, 210)-net over F25, using
- net defined by OOA [i] based on linear OOA(2514, 210, F25, 7, 7) (dual of [(210, 7), 1456, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2514, 631, F25, 7) (dual of [631, 617, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(2513, 626, F25, 7) (dual of [626, 613, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 626 | 254−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(259, 626, F25, 5) (dual of [626, 617, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 626 | 254−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(251, 5, F25, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(251, s, F25, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(2514, 631, F25, 7) (dual of [631, 617, 8]-code), using
- net defined by OOA [i] based on linear OOA(2514, 210, F25, 7, 7) (dual of [(210, 7), 1456, 8]-NRT-code), using
- digital (56, 71, 1198371)-net over F25, using
- net defined by OOA [i] based on linear OOA(2571, 1198371, F25, 15, 15) (dual of [(1198371, 15), 17975494, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2571, 8388598, F25, 15) (dual of [8388598, 8388527, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2571, large, F25, 15) (dual of [large, large−71, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2571, large, F25, 15) (dual of [large, large−71, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2571, 8388598, F25, 15) (dual of [8388598, 8388527, 16]-code), using
- net defined by OOA [i] based on linear OOA(2571, 1198371, F25, 15, 15) (dual of [(1198371, 15), 17975494, 16]-NRT-code), using
- digital (7, 14, 210)-net over F25, using
(85−15, 85, large)-Net over F25 — Digital
Digital (70, 85, large)-net over F25, using
- t-expansion [i] based on digital (69, 85, large)-net over F25, using
- 2 times m-reduction [i] based on digital (69, 87, large)-net over F25, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2587, large, F25, 18) (dual of [large, large−87, 19]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2586, large, F25, 18) (dual of [large, large−86, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 1 times code embedding in larger space [i] based on linear OA(2586, large, F25, 18) (dual of [large, large−86, 19]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2587, large, F25, 18) (dual of [large, large−87, 19]-code), using
- 2 times m-reduction [i] based on digital (69, 87, large)-net over F25, using
(85−15, 85, large)-Net in Base 25 — Upper bound on s
There is no (70, 85, large)-net in base 25, because
- 13 times m-reduction [i] would yield (70, 72, large)-net in base 25, but