Best Known (90, 109, s)-Nets in Base 25
(90, 109, 932223)-Net over F25 — Constructive and digital
Digital (90, 109, 932223)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (9, 18, 157)-net over F25, using
- net defined by OOA [i] based on linear OOA(2518, 157, F25, 9, 9) (dual of [(157, 9), 1395, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2518, 629, F25, 9) (dual of [629, 611, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(2518, 631, F25, 9) (dual of [631, 613, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(2517, 626, F25, 9) (dual of [626, 609, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 626 | 254−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- 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(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([0,4]) ⊂ C([0,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2518, 631, F25, 9) (dual of [631, 613, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2518, 629, F25, 9) (dual of [629, 611, 10]-code), using
- net defined by OOA [i] based on linear OOA(2518, 157, F25, 9, 9) (dual of [(157, 9), 1395, 10]-NRT-code), using
- digital (72, 91, 932066)-net over F25, using
- net defined by OOA [i] based on linear OOA(2591, 932066, F25, 19, 19) (dual of [(932066, 19), 17709163, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2591, 8388595, F25, 19) (dual of [8388595, 8388504, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2591, large, F25, 19) (dual of [large, large−91, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2591, large, F25, 19) (dual of [large, large−91, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2591, 8388595, F25, 19) (dual of [8388595, 8388504, 20]-code), using
- net defined by OOA [i] based on linear OOA(2591, 932066, F25, 19, 19) (dual of [(932066, 19), 17709163, 20]-NRT-code), using
- digital (9, 18, 157)-net over F25, using
(90, 109, large)-Net over F25 — Digital
Digital (90, 109, large)-net over F25, using
- 252 times duplication [i] based on digital (88, 107, large)-net over F25, using
- t-expansion [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
- t-expansion [i] based on digital (85, 107, large)-net over F25, using
(90, 109, large)-Net in Base 25 — Upper bound on s
There is no (90, 109, large)-net in base 25, because
- 17 times m-reduction [i] would yield (90, 92, large)-net in base 25, but