Best Known (57, 63, s)-Nets in Base 5
(57, 63, 5636278)-Net over F5 — Constructive and digital
Digital (57, 63, 5636278)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (8, 11, 43876)-net over F5, using
- net defined by OOA [i] based on linear OOA(511, 43876, F5, 3, 3) (dual of [(43876, 3), 131617, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(511, 43876, F5, 2, 3) (dual of [(43876, 2), 87741, 4]-NRT-code), using
- OAs with strength 3, b ≠ 2, and m > 3 are always embeddable [i] based on linear OA(511, 43876, F5, 3) (dual of [43876, 43865, 4]-code or 43876-cap in PG(10,5)), using
- appending kth column [i] based on linear OOA(511, 43876, F5, 2, 3) (dual of [(43876, 2), 87741, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(511, 43876, F5, 3, 3) (dual of [(43876, 3), 131617, 4]-NRT-code), using
- digital (46, 52, 5592402)-net over F5, using
- trace code for nets [i] 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
- trace code for nets [i] based on digital (20, 26, 2796201)-net over F25, using
- digital (8, 11, 43876)-net over F5, using
(57, 63, large)-Net over F5 — Digital
Digital (57, 63, large)-net over F5, using
- 51 times duplication [i] based on digital (56, 62, large)-net over F5, using
- t-expansion [i] based on digital (54, 62, large)-net over F5, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(562, large, F5, 8) (dual of [large, large−62, 9]-code), using
- 1 times code embedding in larger space [i] based on linear OA(561, large, F5, 8) (dual of [large, large−61, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- 1 times code embedding in larger space [i] based on linear OA(561, large, F5, 8) (dual of [large, large−61, 9]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(562, large, F5, 8) (dual of [large, large−62, 9]-code), using
- t-expansion [i] based on digital (54, 62, large)-net over F5, using
(57, 63, large)-Net in Base 5 — Upper bound on s
There is no (57, 63, large)-net in base 5, because
- 4 times m-reduction [i] would yield (57, 59, large)-net in base 5, but