Best Known (64, 81, s)-Nets in Base 81
(64, 81, 1050216)-Net over F81 — Constructive and digital
Digital (64, 81, 1050216)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (8, 16, 1641)-net over F81, using
- net defined by OOA [i] based on linear OOA(8116, 1641, F81, 8, 8) (dual of [(1641, 8), 13112, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(8116, 6564, F81, 8) (dual of [6564, 6548, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(8116, 6566, F81, 8) (dual of [6566, 6550, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] 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]
- linear OA(8111, 6561, F81, 6) (dual of [6561, 6550, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(8116, 6566, F81, 8) (dual of [6566, 6550, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(8116, 6564, F81, 8) (dual of [6564, 6548, 9]-code), using
- net defined by OOA [i] based on linear OOA(8116, 1641, F81, 8, 8) (dual of [(1641, 8), 13112, 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 (8, 16, 1641)-net over F81, using
(64, 81, large)-Net over F81 — Digital
Digital (64, 81, large)-net over F81, using
- t-expansion [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
(64, 81, large)-Net in Base 81 — Upper bound on s
There is no (64, 81, large)-net in base 81, because
- 15 times m-reduction [i] would yield (64, 66, large)-net in base 81, but