Best Known (106−13, 106, s)-Nets in Base 5
(106−13, 106, 1398100)-Net over F5 — Constructive and digital
Digital (93, 106, 1398100)-net over F5, using
- 55 times duplication [i] based on digital (88, 101, 1398100)-net over F5, using
- net defined by OOA [i] based on linear OOA(5101, 1398100, F5, 13, 13) (dual of [(1398100, 13), 18175199, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(5101, 8388601, F5, 13) (dual of [8388601, 8388500, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(5101, large, F5, 13) (dual of [large, large−101, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 520−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(5101, large, F5, 13) (dual of [large, large−101, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(5101, 8388601, F5, 13) (dual of [8388601, 8388500, 14]-code), using
- net defined by OOA [i] based on linear OOA(5101, 1398100, F5, 13, 13) (dual of [(1398100, 13), 18175199, 14]-NRT-code), using
(106−13, 106, 5766868)-Net over F5 — Digital
Digital (93, 106, 5766868)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5106, 5766868, F5, 13) (dual of [5766868, 5766762, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(5106, large, F5, 13) (dual of [large, large−106, 14]-code), using
- 5 times code embedding in larger space [i] based on linear OA(5101, large, F5, 13) (dual of [large, large−101, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 520−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- 5 times code embedding in larger space [i] based on linear OA(5101, large, F5, 13) (dual of [large, large−101, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(5106, large, F5, 13) (dual of [large, large−106, 14]-code), using
(106−13, 106, large)-Net in Base 5 — Upper bound on s
There is no (93, 106, large)-net in base 5, because
- 11 times m-reduction [i] would yield (93, 95, large)-net in base 5, but