Best Known (54, 54+8, s)-Nets in Base 5
(54, 54+8, 2097150)-Net over F5 — Constructive and digital
Digital (54, 62, 2097150)-net over F5, using
- 51 times duplication [i] based on digital (53, 61, 2097150)-net over F5, using
- net defined by OOA [i] based on linear OOA(561, 2097150, F5, 8, 8) (dual of [(2097150, 8), 16777139, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(561, 8388600, F5, 8) (dual of [8388600, 8388539, 9]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(561, large, F5, 8) (dual of [large, large−61, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(561, 8388600, F5, 8) (dual of [8388600, 8388539, 9]-code), using
- net defined by OOA [i] based on linear OOA(561, 2097150, F5, 8, 8) (dual of [(2097150, 8), 16777139, 9]-NRT-code), using
(54, 54+8, large)-Net over F5 — Digital
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
(54, 54+8, large)-Net in Base 5 — Upper bound on s
There is no (54, 62, large)-net in base 5, because
- 6 times m-reduction [i] would yield (54, 56, large)-net in base 5, but