Best Known (37, 37+9, s)-Nets in Base 81
(37, 37+9, 4194300)-Net over F81 — Constructive and digital
Digital (37, 46, 4194300)-net over F81, using
- net defined by OOA [i] based on linear OOA(8146, 4194300, F81, 10, 9) (dual of [(4194300, 10), 41942954, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(8146, 8388601, F81, 2, 9) (dual of [(8388601, 2), 16777156, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8146, 8388602, F81, 2, 9) (dual of [(8388602, 2), 16777158, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(8113, 4194301, F81, 2, 4) (dual of [(4194301, 2), 8388589, 5]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8113, 8388602, F81, 4) (dual of [8388602, 8388589, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(8113, large, F81, 4) (dual of [large, large−13, 5]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(8113, large, F81, 4) (dual of [large, large−13, 5]-code), using
- OOA 2-folding [i] based on linear OA(8113, 8388602, F81, 4) (dual of [8388602, 8388589, 5]-code), using
- linear OOA(8133, 4194301, F81, 2, 9) (dual of [(4194301, 2), 8388569, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8133, 8388602, F81, 9) (dual of [8388602, 8388569, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(8133, large, F81, 9) (dual of [large, large−33, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, 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(8133, large, F81, 9) (dual of [large, large−33, 10]-code), using
- OOA 2-folding [i] based on linear OA(8133, 8388602, F81, 9) (dual of [8388602, 8388569, 10]-code), using
- linear OOA(8113, 4194301, F81, 2, 4) (dual of [(4194301, 2), 8388589, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OOA(8146, 8388602, F81, 2, 9) (dual of [(8388602, 2), 16777158, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(8146, 8388601, F81, 2, 9) (dual of [(8388601, 2), 16777156, 10]-NRT-code), using
(37, 37+9, large)-Net over F81 — Digital
Digital (37, 46, large)-net over F81, using
- t-expansion [i] based on digital (36, 46, large)-net over F81, using
- 3 times m-reduction [i] based on digital (36, 49, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8149, large, F81, 13) (dual of [large, large−49, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, 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(8149, large, F81, 13) (dual of [large, large−49, 14]-code), using
- 3 times m-reduction [i] based on digital (36, 49, large)-net over F81, using
(37, 37+9, large)-Net in Base 81 — Upper bound on s
There is no (37, 46, large)-net in base 81, because
- 7 times m-reduction [i] would yield (37, 39, large)-net in base 81, but