Best Known (123−12, 123, s)-Nets in Base 3
(123−12, 123, 1398100)-Net over F3 — Constructive and digital
Digital (111, 123, 1398100)-net over F3, using
- 32 times duplication [i] based on digital (109, 121, 1398100)-net over F3, using
- t-expansion [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
- t-expansion [i] based on digital (108, 121, 1398100)-net over F3, using
(123−12, 123, 4194302)-Net over F3 — Digital
Digital (111, 123, 4194302)-net over F3, using
- 31 times duplication [i] based on digital (110, 122, 4194302)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3122, 4194302, F3, 2, 12) (dual of [(4194302, 2), 8388482, 13]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3120, 4194301, F3, 2, 12) (dual of [(4194301, 2), 8388482, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3120, 8388602, F3, 12) (dual of [8388602, 8388482, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(3120, large, F3, 12) (dual of [large, large−120, 13]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(3120, large, F3, 12) (dual of [large, large−120, 13]-code), using
- OOA 2-folding [i] based on linear OA(3120, 8388602, F3, 12) (dual of [8388602, 8388482, 13]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3120, 4194301, F3, 2, 12) (dual of [(4194301, 2), 8388482, 13]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3122, 4194302, F3, 2, 12) (dual of [(4194302, 2), 8388482, 13]-NRT-code), using
(123−12, 123, large)-Net in Base 3 — Upper bound on s
There is no (111, 123, large)-net in base 3, because
- 10 times m-reduction [i] would yield (111, 113, large)-net in base 3, but