Best Known (213−19, 213, s)-Nets in Base 3
(213−19, 213, 932180)-Net over F3 — Constructive and digital
Digital (194, 213, 932180)-net over F3, using
- 32 times duplication [i] based on digital (192, 211, 932180)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (21, 30, 114)-net over F3, using
- trace code for nets [i] based on digital (1, 10, 38)-net over F27, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 1 and N(F) ≥ 38, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- trace code for nets [i] based on digital (1, 10, 38)-net over F27, using
- digital (162, 181, 932066)-net over F3, using
- net defined by OOA [i] based on linear OOA(3181, 932066, F3, 19, 19) (dual of [(932066, 19), 17709073, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3181, 8388595, F3, 19) (dual of [8388595, 8388414, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3181, large, F3, 19) (dual of [large, large−181, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 330−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(3181, large, F3, 19) (dual of [large, large−181, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3181, 8388595, F3, 19) (dual of [8388595, 8388414, 20]-code), using
- net defined by OOA [i] based on linear OOA(3181, 932066, F3, 19, 19) (dual of [(932066, 19), 17709073, 20]-NRT-code), using
- digital (21, 30, 114)-net over F3, using
- (u, u+v)-construction [i] based on
(213−19, 213, 4194515)-Net over F3 — Digital
Digital (194, 213, 4194515)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3213, 4194515, F3, 2, 19) (dual of [(4194515, 2), 8388817, 20]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(332, 214, F3, 2, 9) (dual of [(214, 2), 396, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(332, 214, F3, 9) (dual of [214, 182, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(332, 255, F3, 9) (dual of [255, 223, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(331, 244, F3, 9) (dual of [244, 213, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 244 | 310−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(321, 244, F3, 7) (dual of [244, 223, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 244 | 310−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(31, 11, F3, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(332, 255, F3, 9) (dual of [255, 223, 10]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(332, 214, F3, 9) (dual of [214, 182, 10]-code), using
- linear OOA(3181, 4194301, F3, 2, 19) (dual of [(4194301, 2), 8388421, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3181, 8388602, F3, 19) (dual of [8388602, 8388421, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3181, large, F3, 19) (dual of [large, large−181, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 330−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(3181, large, F3, 19) (dual of [large, large−181, 20]-code), using
- OOA 2-folding [i] based on linear OA(3181, 8388602, F3, 19) (dual of [8388602, 8388421, 20]-code), using
- linear OOA(332, 214, F3, 2, 9) (dual of [(214, 2), 396, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
(213−19, 213, large)-Net in Base 3 — Upper bound on s
There is no (194, 213, large)-net in base 3, because
- 17 times m-reduction [i] would yield (194, 196, large)-net in base 3, but