Best Known (231, 231+7, s)-Nets in Base 3
(231, 231+7, large)-Net over F3 — Constructive and digital
Digital (231, 238, large)-net over F3, using
- 5 times m-reduction [i] based on digital (231, 243, large)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (57, 63, 2796205)-net over F3, using
- net defined by OOA [i] based on linear OOA(363, 2796205, F3, 6, 6) (dual of [(2796205, 6), 16777167, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(363, 2796205, F3, 5, 6) (dual of [(2796205, 5), 13980962, 7]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(33, 4, F3, 5, 3) (dual of [(4, 5), 17, 4]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(5;17,3) [i]
- linear OOA(360, 2796201, F3, 5, 6) (dual of [(2796201, 5), 13980945, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(360, large, F3, 6) (dual of [large, large−60, 7]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OA 3-folding and stacking [i] based on linear OA(360, large, F3, 6) (dual of [large, large−60, 7]-code), using
- linear OOA(33, 4, F3, 5, 3) (dual of [(4, 5), 17, 4]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(363, 2796205, F3, 5, 6) (dual of [(2796205, 5), 13980962, 7]-NRT-code), using
- net defined by OOA [i] based on linear OOA(363, 2796205, F3, 6, 6) (dual of [(2796205, 6), 16777167, 7]-NRT-code), using
- digital (168, 180, 5592400)-net over F3, using
- trace code for nets [i] based on digital (78, 90, 2796200)-net over F9, using
- net defined by OOA [i] based on linear OOA(990, 2796200, F9, 14, 12) (dual of [(2796200, 14), 39146710, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(990, 8388601, F9, 2, 12) (dual of [(8388601, 2), 16777112, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(990, 8388602, F9, 2, 12) (dual of [(8388602, 2), 16777114, 13]-NRT-code), using
- trace code [i] based on linear OOA(8145, 4194301, F81, 2, 12) (dual of [(4194301, 2), 8388557, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8145, 8388602, F81, 12) (dual of [8388602, 8388557, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(8145, large, F81, 12) (dual of [large, large−45, 13]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(8145, large, F81, 12) (dual of [large, large−45, 13]-code), using
- OOA 2-folding [i] based on linear OA(8145, 8388602, F81, 12) (dual of [8388602, 8388557, 13]-code), using
- trace code [i] based on linear OOA(8145, 4194301, F81, 2, 12) (dual of [(4194301, 2), 8388557, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(990, 8388602, F9, 2, 12) (dual of [(8388602, 2), 16777114, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(990, 8388601, F9, 2, 12) (dual of [(8388601, 2), 16777112, 13]-NRT-code), using
- net defined by OOA [i] based on linear OOA(990, 2796200, F9, 14, 12) (dual of [(2796200, 14), 39146710, 13]-NRT-code), using
- trace code for nets [i] based on digital (78, 90, 2796200)-net over F9, using
- digital (57, 63, 2796205)-net over F3, using
- (u, u+v)-construction [i] based on
(231, 231+7, large)-Net in Base 3 — Upper bound on s
There is no (231, 238, large)-net in base 3, because
- 5 times m-reduction [i] would yield (231, 233, large)-net in base 3, but