Best Known (56, 56+9, s)-Nets in Base 27
(56, 56+9, 5439433)-Net over F27 — Constructive and digital
Digital (56, 65, 5439433)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (5, 8, 1245133)-net over F27, using
- net defined by OOA [i] based on linear OOA(278, 1245133, F27, 3, 3) (dual of [(1245133, 3), 3735391, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(278, 1245133, F27, 2, 3) (dual of [(1245133, 2), 2490258, 4]-NRT-code), using
- OAs with strength 3, b ≠ 2, and m > 3 are always embeddable [i] based on linear OA(278, 1245133, F27, 3) (dual of [1245133, 1245125, 4]-code or 1245133-cap in PG(7,27)), using
- appending kth column [i] based on linear OOA(278, 1245133, F27, 2, 3) (dual of [(1245133, 2), 2490258, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(278, 1245133, F27, 3, 3) (dual of [(1245133, 3), 3735391, 4]-NRT-code), using
- digital (12, 16, 2097150)-net over F27, using
- s-reduction based on digital (12, 16, 4194301)-net over F27, using
- net defined by OOA [i] based on linear OOA(2716, 4194301, F27, 4, 4) (dual of [(4194301, 4), 16777188, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(2716, 8388602, F27, 4) (dual of [8388602, 8388586, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(2716, large, F27, 4) (dual of [large, large−16, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(2716, large, F27, 4) (dual of [large, large−16, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(2716, 8388602, F27, 4) (dual of [8388602, 8388586, 5]-code), using
- net defined by OOA [i] based on linear OOA(2716, 4194301, F27, 4, 4) (dual of [(4194301, 4), 16777188, 5]-NRT-code), using
- s-reduction based on digital (12, 16, 4194301)-net over F27, using
- digital (32, 41, 2097150)-net over F27, using
- net defined by OOA [i] based on linear OOA(2741, 2097150, F27, 9, 9) (dual of [(2097150, 9), 18874309, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2741, 8388601, F27, 9) (dual of [8388601, 8388560, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(2741, large, F27, 9) (dual of [large, large−41, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2741, large, F27, 9) (dual of [large, large−41, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2741, 8388601, F27, 9) (dual of [8388601, 8388560, 10]-code), using
- net defined by OOA [i] based on linear OOA(2741, 2097150, F27, 9, 9) (dual of [(2097150, 9), 18874309, 10]-NRT-code), using
- digital (5, 8, 1245133)-net over F27, using
(56, 56+9, large)-Net over F27 — Digital
Digital (56, 65, large)-net over F27, using
- 6 times m-reduction [i] based on digital (56, 71, large)-net over F27, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2771, large, F27, 15) (dual of [large, large−71, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2771, large, F27, 15) (dual of [large, large−71, 16]-code), using
(56, 56+9, large)-Net in Base 27 — Upper bound on s
There is no (56, 65, large)-net in base 27, because
- 7 times m-reduction [i] would yield (56, 58, large)-net in base 27, but