Best Known (83, 83+12, s)-Nets in Base 25
(83, 83+12, 2991514)-Net over F25 — Constructive and digital
Digital (83, 95, 2991514)-net over F25, using
- generalized (u, u+v)-construction [i] based on
- digital (9, 13, 195314)-net over F25, using
- net defined by OOA [i] based on linear OOA(2513, 195314, F25, 4, 4) (dual of [(195314, 4), 781243, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(2513, 390628, F25, 4) (dual of [390628, 390615, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(2513, 390629, F25, 4) (dual of [390629, 390616, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(2513, 390625, F25, 4) (dual of [390625, 390612, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(259, 390625, F25, 3) (dual of [390625, 390616, 4]-code or 390625-cap in PG(8,25)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(250, 4, F25, 0) (dual of [4, 4, 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
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(2513, 390629, F25, 4) (dual of [390629, 390616, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(2513, 390628, F25, 4) (dual of [390628, 390615, 5]-code), using
- net defined by OOA [i] based on linear OOA(2513, 195314, F25, 4, 4) (dual of [(195314, 4), 781243, 5]-NRT-code), using
- digital (20, 26, 1398100)-net over F25, using
- s-reduction based on digital (20, 26, 2796201)-net over F25, using
- net defined by OOA [i] based on linear OOA(2526, 2796201, F25, 6, 6) (dual of [(2796201, 6), 16777180, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(2526, large, F25, 6) (dual of [large, large−26, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OA 3-folding and stacking [i] based on linear OA(2526, large, F25, 6) (dual of [large, large−26, 7]-code), using
- net defined by OOA [i] based on linear OOA(2526, 2796201, F25, 6, 6) (dual of [(2796201, 6), 16777180, 7]-NRT-code), using
- s-reduction based on digital (20, 26, 2796201)-net over F25, using
- digital (44, 56, 1398100)-net over F25, using
- net defined by OOA [i] based on linear OOA(2556, 1398100, F25, 12, 12) (dual of [(1398100, 12), 16777144, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(2556, 8388600, F25, 12) (dual of [8388600, 8388544, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(2556, large, F25, 12) (dual of [large, large−56, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(2556, large, F25, 12) (dual of [large, large−56, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(2556, 8388600, F25, 12) (dual of [8388600, 8388544, 13]-code), using
- net defined by OOA [i] based on linear OOA(2556, 1398100, F25, 12, 12) (dual of [(1398100, 12), 16777144, 13]-NRT-code), using
- digital (9, 13, 195314)-net over F25, using
(83, 83+12, large)-Net over F25 — Digital
Digital (83, 95, large)-net over F25, using
- t-expansion [i] based on digital (81, 95, large)-net over F25, using
- 7 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
- 7 times m-reduction [i] based on digital (81, 102, large)-net over F25, using
(83, 83+12, large)-Net in Base 25 — Upper bound on s
There is no (83, 95, large)-net in base 25, because
- 10 times m-reduction [i] would yield (83, 85, large)-net in base 25, but