Best Known (219, 219+24, s)-Nets in Base 3
(219, 219+24, 699050)-Net over F3 — Constructive and digital
Digital (219, 243, 699050)-net over F3, using
- 32 times duplication [i] based on digital (217, 241, 699050)-net over F3, using
- t-expansion [i] based on digital (216, 241, 699050)-net over F3, using
- net defined by OOA [i] based on linear OOA(3241, 699050, F3, 25, 25) (dual of [(699050, 25), 17476009, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3241, 8388601, F3, 25) (dual of [8388601, 8388360, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3241, large, F3, 25) (dual of [large, large−241, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 330−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(3241, large, F3, 25) (dual of [large, large−241, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3241, 8388601, F3, 25) (dual of [8388601, 8388360, 26]-code), using
- net defined by OOA [i] based on linear OOA(3241, 699050, F3, 25, 25) (dual of [(699050, 25), 17476009, 26]-NRT-code), using
- t-expansion [i] based on digital (216, 241, 699050)-net over F3, using
(219, 219+24, 2206620)-Net over F3 — Digital
Digital (219, 243, 2206620)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3243, 2206620, F3, 3, 24) (dual of [(2206620, 3), 6619617, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3243, 2796202, F3, 3, 24) (dual of [(2796202, 3), 8388363, 25]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3240, 2796201, F3, 3, 24) (dual of [(2796201, 3), 8388363, 25]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3240, large, F3, 24) (dual of [large, large−240, 25]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- OOA 3-folding [i] based on linear OA(3240, large, F3, 24) (dual of [large, large−240, 25]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3240, 2796201, F3, 3, 24) (dual of [(2796201, 3), 8388363, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3243, 2796202, F3, 3, 24) (dual of [(2796202, 3), 8388363, 25]-NRT-code), using
(219, 219+24, large)-Net in Base 3 — Upper bound on s
There is no (219, 243, large)-net in base 3, because
- 22 times m-reduction [i] would yield (219, 221, large)-net in base 3, but