Best Known (123−13, 123, s)-Nets in Base 3
(123−13, 123, 1398100)-Net over F3 — Constructive and digital
Digital (110, 123, 1398100)-net over F3, using
- 32 times duplication [i] based on digital (108, 121, 1398100)-net over F3, using
- net defined by OOA [i] based on linear OOA(3121, 1398100, F3, 13, 13) (dual of [(1398100, 13), 18175179, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3121, 8388601, F3, 13) (dual of [8388601, 8388480, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3121, large, F3, 13) (dual of [large, large−121, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 330−1, 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(3121, large, F3, 13) (dual of [large, large−121, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3121, 8388601, F3, 13) (dual of [8388601, 8388480, 14]-code), using
- net defined by OOA [i] based on linear OOA(3121, 1398100, F3, 13, 13) (dual of [(1398100, 13), 18175179, 14]-NRT-code), using
(123−13, 123, 2796201)-Net over F3 — Digital
Digital (110, 123, 2796201)-net over F3, using
- 32 times duplication [i] based on digital (108, 121, 2796201)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3121, 2796201, F3, 3, 13) (dual of [(2796201, 3), 8388482, 14]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3121, large, F3, 13) (dual of [large, large−121, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 330−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- OOA 3-folding [i] based on linear OA(3121, large, F3, 13) (dual of [large, large−121, 14]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3121, 2796201, F3, 3, 13) (dual of [(2796201, 3), 8388482, 14]-NRT-code), using
(123−13, 123, large)-Net in Base 3 — Upper bound on s
There is no (110, 123, large)-net in base 3, because
- 11 times m-reduction [i] would yield (110, 112, large)-net in base 3, but