Best Known (52, 65, s)-Nets in Base 81
(52, 65, 1575248)-Net over F81 — Constructive and digital
Digital (52, 65, 1575248)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (10, 16, 177148)-net over F81, using
- net defined by OOA [i] based on linear OOA(8116, 177148, F81, 6, 6) (dual of [(177148, 6), 1062872, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(8116, 531444, F81, 6) (dual of [531444, 531428, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(8116, 531441, F81, 6) (dual of [531441, 531425, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(8113, 531441, F81, 5) (dual of [531441, 531428, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(810, 3, F81, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- OA 3-folding and stacking [i] based on linear OA(8116, 531444, F81, 6) (dual of [531444, 531428, 7]-code), using
- net defined by OOA [i] based on linear OOA(8116, 177148, F81, 6, 6) (dual of [(177148, 6), 1062872, 7]-NRT-code), using
- digital (36, 49, 1398100)-net over F81, using
- net defined by OOA [i] based on linear OOA(8149, 1398100, F81, 13, 13) (dual of [(1398100, 13), 18175251, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(8149, 8388601, F81, 13) (dual of [8388601, 8388552, 14]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(8149, large, F81, 13) (dual of [large, large−49, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(8149, 8388601, F81, 13) (dual of [8388601, 8388552, 14]-code), using
- net defined by OOA [i] based on linear OOA(8149, 1398100, F81, 13, 13) (dual of [(1398100, 13), 18175251, 14]-NRT-code), using
- digital (10, 16, 177148)-net over F81, using
(52, 65, large)-Net over F81 — Digital
Digital (52, 65, large)-net over F81, using
- t-expansion [i] based on digital (51, 65, large)-net over F81, using
- 4 times m-reduction [i] based on digital (51, 69, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8169, large, F81, 18) (dual of [large, large−69, 19]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8169, large, F81, 18) (dual of [large, large−69, 19]-code), using
- 4 times m-reduction [i] based on digital (51, 69, large)-net over F81, using
(52, 65, large)-Net in Base 81 — Upper bound on s
There is no (52, 65, large)-net in base 81, because
- 11 times m-reduction [i] would yield (52, 54, large)-net in base 81, but