Best Known (81, 81+8, s)-Nets in Base 5
(81, 81+8, 4195868)-Net over F5 — Constructive and digital
Digital (81, 89, 4195868)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (13, 17, 1568)-net over F5, using
- net defined by OOA [i] based on linear OOA(517, 1568, F5, 4, 4) (dual of [(1568, 4), 6255, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(517, 1568, F5, 3, 4) (dual of [(1568, 3), 4687, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(517, 3136, F5, 4) (dual of [3136, 3119, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(1) [i] based on
- linear OA(516, 3125, F5, 4) (dual of [3125, 3109, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(56, 3125, F5, 2) (dual of [3125, 3119, 3]-code), using an extension Ce(1) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,1], and designed minimum distance d ≥ |I|+1 = 2 [i]
- linear OA(51, 11, F5, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(1) [i] based on
- OA 2-folding and stacking [i] based on linear OA(517, 3136, F5, 4) (dual of [3136, 3119, 5]-code), using
- appending kth column [i] based on linear OOA(517, 1568, F5, 3, 4) (dual of [(1568, 3), 4687, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(517, 1568, F5, 4, 4) (dual of [(1568, 4), 6255, 5]-NRT-code), using
- digital (64, 72, 4194300)-net over F5, using
- net defined by OOA [i] based on linear OOA(572, 4194300, F5, 10, 8) (dual of [(4194300, 10), 41942928, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(572, 8388601, F5, 2, 8) (dual of [(8388601, 2), 16777130, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(572, 8388602, F5, 2, 8) (dual of [(8388602, 2), 16777132, 9]-NRT-code), using
- trace code [i] based on linear OOA(2536, 4194301, F25, 2, 8) (dual of [(4194301, 2), 8388566, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2536, 8388602, F25, 8) (dual of [8388602, 8388566, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(2536, large, F25, 8) (dual of [large, large−36, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(2536, large, F25, 8) (dual of [large, large−36, 9]-code), using
- OOA 2-folding [i] based on linear OA(2536, 8388602, F25, 8) (dual of [8388602, 8388566, 9]-code), using
- trace code [i] based on linear OOA(2536, 4194301, F25, 2, 8) (dual of [(4194301, 2), 8388566, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(572, 8388602, F5, 2, 8) (dual of [(8388602, 2), 16777132, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(572, 8388601, F5, 2, 8) (dual of [(8388601, 2), 16777130, 9]-NRT-code), using
- net defined by OOA [i] based on linear OOA(572, 4194300, F5, 10, 8) (dual of [(4194300, 10), 41942928, 9]-NRT-code), using
- digital (13, 17, 1568)-net over F5, using
(81, 81+8, large)-Net over F5 — Digital
Digital (81, 89, large)-net over F5, using
- t-expansion [i] based on digital (79, 89, large)-net over F5, using
- 1 times m-reduction [i] based on digital (79, 90, large)-net over F5, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(590, large, F5, 11) (dual of [large, large−90, 12]-code), using
- 9 times code embedding in larger space [i] based on linear OA(581, large, F5, 11) (dual of [large, large−81, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 520−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- 9 times code embedding in larger space [i] based on linear OA(581, large, F5, 11) (dual of [large, large−81, 12]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(590, large, F5, 11) (dual of [large, large−90, 12]-code), using
- 1 times m-reduction [i] based on digital (79, 90, large)-net over F5, using
(81, 81+8, large)-Net in Base 5 — Upper bound on s
There is no (81, 89, large)-net in base 5, because
- 6 times m-reduction [i] would yield (81, 83, large)-net in base 5, but