Best Known (214−19, 214, s)-Nets in Base 3
(214−19, 214, 932210)-Net over F3 — Constructive and digital
Digital (195, 214, 932210)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (24, 33, 144)-net over F3, using
- trace code for nets [i] based on digital (2, 11, 48)-net over F27, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 2 and N(F) ≥ 48, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
- trace code for nets [i] based on digital (2, 11, 48)-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 (24, 33, 144)-net over F3, using
(214−19, 214, 4194552)-Net over F3 — Digital
Digital (195, 214, 4194552)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3214, 4194552, F3, 2, 19) (dual of [(4194552, 2), 8388890, 20]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(333, 251, F3, 2, 9) (dual of [(251, 2), 469, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(333, 251, F3, 9) (dual of [251, 218, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(333, 259, F3, 9) (dual of [259, 226, 10]-code), using
- construction XX applied to C1 = C([239,4]), C2 = C([0,6]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([239,6]) [i] based on
- linear OA(326, 242, F3, 8) (dual of [242, 216, 9]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {−3,−2,…,4}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(321, 242, F3, 7) (dual of [242, 221, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(331, 242, F3, 10) (dual of [242, 211, 11]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {−3,−2,…,6}, and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(316, 242, F3, 5) (dual of [242, 226, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [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
- linear OA(31, 6, F3, 1) (dual of [6, 5, 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 (see above)
- construction XX applied to C1 = C([239,4]), C2 = C([0,6]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([239,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(333, 259, F3, 9) (dual of [259, 226, 10]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(333, 251, F3, 9) (dual of [251, 218, 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(333, 251, F3, 2, 9) (dual of [(251, 2), 469, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
(214−19, 214, large)-Net in Base 3 — Upper bound on s
There is no (195, 214, large)-net in base 3, because
- 17 times m-reduction [i] would yield (195, 197, large)-net in base 3, but