Best Known (54, 54+7, s)-Nets in Base 5
(54, 54+7, 2813324)-Net over F5 — Constructive and digital
Digital (54, 61, 2813324)-net over F5, using
- net defined by OOA [i] based on linear OOA(561, 2813324, F5, 9, 7) (dual of [(2813324, 9), 25319855, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(561, 2813325, F5, 3, 7) (dual of [(2813325, 3), 8439914, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(510, 17124, F5, 3, 3) (dual of [(17124, 3), 51362, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(510, 17124, F5, 2, 3) (dual of [(17124, 2), 34238, 4]-NRT-code), using
- linear OOA(551, 2796201, F5, 3, 7) (dual of [(2796201, 3), 8388552, 8]-NRT-code), using
- OOA 3-folding [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]
- OOA 3-folding [i] based on linear OA(551, large, F5, 7) (dual of [large, large−51, 8]-code), using
- linear OOA(510, 17124, F5, 3, 3) (dual of [(17124, 3), 51362, 4]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(561, 2813325, F5, 3, 7) (dual of [(2813325, 3), 8439914, 8]-NRT-code), using
(54, 54+7, large)-Net over F5 — Digital
Digital (54, 61, large)-net over F5, using
- 1 times m-reduction [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
(54, 54+7, large)-Net in Base 5 — Upper bound on s
There is no (54, 61, large)-net in base 5, because
- 5 times m-reduction [i] would yield (54, 56, large)-net in base 5, but