Best Known (50, 50+6, s)-Nets in Base 5
(50, 50+6, 5592428)-Net over F5 — Constructive and digital
Digital (50, 56, 5592428)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (1, 4, 26)-net over F5, using
- net defined by OOA [i] based on linear OOA(54, 26, F5, 3, 3) (dual of [(26, 3), 74, 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 (1, 4, 26)-net over F5, using
(50, 50+6, large)-Net over F5 — Digital
Digital (50, 56, large)-net over F5, using
- 54 times duplication [i] based on digital (46, 52, large)-net over F5, using
- t-expansion [i] based on digital (45, 52, large)-net over F5, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(552, large, F5, 7) (dual of [large, large−52, 8]-code), using
- 1 times code embedding in larger space [i] based on linear OA(551, large, F5, 7) (dual of [large, large−51, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- 1 times code embedding in larger space [i] based on linear OA(551, large, F5, 7) (dual of [large, large−51, 8]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(552, large, F5, 7) (dual of [large, large−52, 8]-code), using
- t-expansion [i] based on digital (45, 52, large)-net over F5, using
(50, 50+6, large)-Net in Base 5 — Upper bound on s
There is no (50, 56, large)-net in base 5, because
- 4 times m-reduction [i] would yield (50, 52, large)-net in base 5, but