Best Known (138−17, 138, s)-Nets in Base 5
(138−17, 138, 1048575)-Net over F5 — Constructive and digital
Digital (121, 138, 1048575)-net over F5, using
- 57 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
(138−17, 138, 4194304)-Net over F5 — Digital
Digital (121, 138, 4194304)-net over F5, using
- 51 times duplication [i] based on digital (120, 137, 4194304)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(5137, 4194304, F5, 2, 17) (dual of [(4194304, 2), 8388471, 18]-NRT-code), using
- 3 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
- 3 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(5137, 4194304, F5, 2, 17) (dual of [(4194304, 2), 8388471, 18]-NRT-code), using
(138−17, 138, large)-Net in Base 5 — Upper bound on s
There is no (121, 138, large)-net in base 5, because
- 15 times m-reduction [i] would yield (121, 123, large)-net in base 5, but