Best Known (231−8, 231, s)-Nets in Base 3
(231−8, 231, large)-Net over F3 — Constructive and digital
Digital (223, 231, large)-net over F3, using
- t-expansion [i] based on digital (221, 231, large)-net over F3, using
- 1 times m-reduction [i] based on digital (221, 232, large)-net over F3, using
- trace code for nets [i] based on digital (47, 58, 2796200)-net over F81, using
- net defined by OOA [i] based on linear OOA(8158, 2796200, F81, 14, 11) (dual of [(2796200, 14), 39146742, 12]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(8158, 8388601, F81, 2, 11) (dual of [(8388601, 2), 16777144, 12]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8158, 8388602, F81, 2, 11) (dual of [(8388602, 2), 16777146, 12]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(8117, 4194301, F81, 2, 5) (dual of [(4194301, 2), 8388585, 6]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8117, 8388602, F81, 5) (dual of [8388602, 8388585, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(8117, large, F81, 5) (dual of [large, large−17, 6]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8117, large, F81, 5) (dual of [large, large−17, 6]-code), using
- OOA 2-folding [i] based on linear OA(8117, 8388602, F81, 5) (dual of [8388602, 8388585, 6]-code), using
- linear OOA(8141, 4194301, F81, 2, 11) (dual of [(4194301, 2), 8388561, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8141, 8388602, F81, 11) (dual of [8388602, 8388561, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(8141, large, F81, 11) (dual of [large, large−41, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8141, large, F81, 11) (dual of [large, large−41, 12]-code), using
- OOA 2-folding [i] based on linear OA(8141, 8388602, F81, 11) (dual of [8388602, 8388561, 12]-code), using
- linear OOA(8117, 4194301, F81, 2, 5) (dual of [(4194301, 2), 8388585, 6]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OOA(8158, 8388602, F81, 2, 11) (dual of [(8388602, 2), 16777146, 12]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(8158, 8388601, F81, 2, 11) (dual of [(8388601, 2), 16777144, 12]-NRT-code), using
- net defined by OOA [i] based on linear OOA(8158, 2796200, F81, 14, 11) (dual of [(2796200, 14), 39146742, 12]-NRT-code), using
- trace code for nets [i] based on digital (47, 58, 2796200)-net over F81, using
- 1 times m-reduction [i] based on digital (221, 232, large)-net over F3, using
(231−8, 231, large)-Net in Base 3 — Upper bound on s
There is no (223, 231, large)-net in base 3, because
- 6 times m-reduction [i] would yield (223, 225, large)-net in base 3, but