Best Known (63, 63+17, s)-Nets in Base 81
(63, 63+17, 1050215)-Net over F81 — Constructive and digital
Digital (63, 80, 1050215)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (7, 15, 1640)-net over F81, using
- net defined by OOA [i] based on linear OOA(8115, 1640, F81, 8, 8) (dual of [(1640, 8), 13105, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(8115, 6560, F81, 8) (dual of [6560, 6545, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(8115, 6561, F81, 8) (dual of [6561, 6546, 9]-code), using
- an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(8115, 6561, F81, 8) (dual of [6561, 6546, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(8115, 6560, F81, 8) (dual of [6560, 6545, 9]-code), using
- net defined by OOA [i] based on linear OOA(8115, 1640, F81, 8, 8) (dual of [(1640, 8), 13105, 9]-NRT-code), using
- digital (48, 65, 1048575)-net over F81, using
- net defined by OOA [i] based on linear OOA(8165, 1048575, F81, 17, 17) (dual of [(1048575, 17), 17825710, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(8165, 8388601, F81, 17) (dual of [8388601, 8388536, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(8165, large, F81, 17) (dual of [large, large−65, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8165, large, F81, 17) (dual of [large, large−65, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(8165, 8388601, F81, 17) (dual of [8388601, 8388536, 18]-code), using
- net defined by OOA [i] based on linear OOA(8165, 1048575, F81, 17, 17) (dual of [(1048575, 17), 17825710, 18]-NRT-code), using
- digital (7, 15, 1640)-net over F81, using
(63, 63+17, large)-Net over F81 — Digital
Digital (63, 80, large)-net over F81, using
- t-expansion [i] based on digital (60, 80, large)-net over F81, using
- 1 times m-reduction [i] based on digital (60, 81, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8181, large, F81, 21) (dual of [large, large−81, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8181, large, F81, 21) (dual of [large, large−81, 22]-code), using
- 1 times m-reduction [i] based on digital (60, 81, large)-net over F81, using
(63, 63+17, large)-Net in Base 81 — Upper bound on s
There is no (63, 80, large)-net in base 81, because
- 15 times m-reduction [i] would yield (63, 65, large)-net in base 81, but