Best Known (117, 117+17, s)-Nets in Base 5
(117, 117+17, 1048575)-Net over F5 — Constructive and digital
Digital (117, 134, 1048575)-net over F5, using
- 53 times duplication [i] based on digital (114, 131, 1048575)-net over F5, using
- net defined by OOA [i] based on linear OOA(5131, 1048575, F5, 17, 17) (dual of [(1048575, 17), 17825644, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(5131, 8388601, F5, 17) (dual of [8388601, 8388470, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(5131, large, F5, 17) (dual of [large, large−131, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(5131, large, F5, 17) (dual of [large, large−131, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(5131, 8388601, F5, 17) (dual of [8388601, 8388470, 18]-code), using
- net defined by OOA [i] based on linear OOA(5131, 1048575, F5, 17, 17) (dual of [(1048575, 17), 17825644, 18]-NRT-code), using
(117, 117+17, 4194302)-Net over F5 — Digital
Digital (117, 134, 4194302)-net over F5, using
- 51 times duplication [i] based on digital (116, 133, 4194302)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(5133, 4194302, F5, 2, 17) (dual of [(4194302, 2), 8388471, 18]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(5131, 4194301, F5, 2, 17) (dual of [(4194301, 2), 8388471, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(5131, 8388602, F5, 17) (dual of [8388602, 8388471, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(5131, large, F5, 17) (dual of [large, large−131, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(5131, large, F5, 17) (dual of [large, large−131, 18]-code), using
- OOA 2-folding [i] based on linear OA(5131, 8388602, F5, 17) (dual of [8388602, 8388471, 18]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(5131, 4194301, F5, 2, 17) (dual of [(4194301, 2), 8388471, 18]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(5133, 4194302, F5, 2, 17) (dual of [(4194302, 2), 8388471, 18]-NRT-code), using
(117, 117+17, large)-Net in Base 5 — Upper bound on s
There is no (117, 134, large)-net in base 5, because
- 15 times m-reduction [i] would yield (117, 119, large)-net in base 5, but