Best Known (50, 59, s)-Nets in Base 27
(50, 59, 4194301)-Net over F27 — Constructive and digital
Digital (50, 59, 4194301)-net over F27, using
- net defined by OOA [i] based on linear OOA(2759, 4194301, F27, 10, 9) (dual of [(4194301, 10), 41942951, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(2759, large, F27, 2, 9), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2757, 8388602, F27, 2, 9) (dual of [(8388602, 2), 16777147, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(2716, 4194301, F27, 2, 4) (dual of [(4194301, 2), 8388586, 5]-NRT-code), using
- OOA 2-folding [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
- OOA 2-folding [i] based on linear OA(2716, 8388602, F27, 4) (dual of [8388602, 8388586, 5]-code), using
- linear OOA(2741, 4194301, F27, 2, 9) (dual of [(4194301, 2), 8388561, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2741, 8388602, F27, 9) (dual of [8388602, 8388561, 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 2-folding [i] based on linear OA(2741, 8388602, F27, 9) (dual of [8388602, 8388561, 10]-code), using
- linear OOA(2716, 4194301, F27, 2, 4) (dual of [(4194301, 2), 8388586, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2757, 8388602, F27, 2, 9) (dual of [(8388602, 2), 16777147, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(2759, large, F27, 2, 9), using
(50, 59, large)-Net over F27 — Digital
Digital (50, 59, large)-net over F27, using
- t-expansion [i] based on digital (48, 59, large)-net over F27, using
- 2 times m-reduction [i] based on digital (48, 61, large)-net over F27, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2761, large, F27, 13) (dual of [large, large−61, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2761, large, F27, 13) (dual of [large, large−61, 14]-code), using
- 2 times m-reduction [i] based on digital (48, 61, large)-net over F27, using
(50, 59, large)-Net in Base 27 — Upper bound on s
There is no (50, 59, large)-net in base 27, because
- 7 times m-reduction [i] would yield (50, 52, large)-net in base 27, but