Best Known (64−10, 64, s)-Nets in Base 25
(64−10, 64, 1873037)-Net over F25 — Constructive and digital
Digital (54, 64, 1873037)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (13, 18, 195317)-net over F25, using
- net defined by OOA [i] based on linear OOA(2518, 195317, F25, 5, 5) (dual of [(195317, 5), 976567, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(2518, 390635, F25, 5) (dual of [390635, 390617, 6]-code), using
- construction X applied to C([0,2]) ⊂ C([0,1]) [i] based on
- linear OA(2517, 390626, F25, 5) (dual of [390626, 390609, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 390626 | 258−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(259, 390626, F25, 3) (dual of [390626, 390617, 4]-code or 390626-cap in PG(8,25)), using the expurgated narrow-sense BCH-code C(I) with length 390626 | 258−1, defining interval I = [0,1], and minimum distance d ≥ |{−1,0,1}|+1 = 4 (BCH-bound) [i]
- linear OA(251, 9, F25, 1) (dual of [9, 8, 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([0,2]) ⊂ C([0,1]) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(2518, 390635, F25, 5) (dual of [390635, 390617, 6]-code), using
- net defined by OOA [i] based on linear OOA(2518, 195317, F25, 5, 5) (dual of [(195317, 5), 976567, 6]-NRT-code), using
- digital (36, 46, 1677720)-net over F25, using
- net defined by OOA [i] based on linear OOA(2546, 1677720, F25, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(2546, 8388600, F25, 10) (dual of [8388600, 8388554, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(2546, large, F25, 10) (dual of [large, large−46, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(2546, large, F25, 10) (dual of [large, large−46, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(2546, 8388600, F25, 10) (dual of [8388600, 8388554, 11]-code), using
- net defined by OOA [i] based on linear OOA(2546, 1677720, F25, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
- digital (13, 18, 195317)-net over F25, using
(64−10, 64, large)-Net over F25 — Digital
Digital (54, 64, large)-net over F25, using
- t-expansion [i] based on digital (53, 64, large)-net over F25, using
- 3 times m-reduction [i] based on digital (53, 67, large)-net over F25, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2567, large, F25, 14) (dual of [large, large−67, 15]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2566, large, F25, 14) (dual of [large, large−66, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- 1 times code embedding in larger space [i] based on linear OA(2566, large, F25, 14) (dual of [large, large−66, 15]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2567, large, F25, 14) (dual of [large, large−67, 15]-code), using
- 3 times m-reduction [i] based on digital (53, 67, large)-net over F25, using
(64−10, 64, large)-Net in Base 25 — Upper bound on s
There is no (54, 64, large)-net in base 25, because
- 8 times m-reduction [i] would yield (54, 56, large)-net in base 25, but