Best Known (81−15, 81, s)-Nets in Base 81
(81−15, 81, 1391836)-Net over F81 — Constructive and digital
Digital (66, 81, 1391836)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (17, 24, 193465)-net over F81, using
- net defined by OOA [i] based on linear OOA(8124, 193465, F81, 9, 7) (dual of [(193465, 9), 1741161, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(8124, 193466, F81, 3, 7) (dual of [(193466, 3), 580374, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(815, 16318, F81, 3, 3) (dual of [(16318, 3), 48949, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(815, 16318, F81, 2, 3) (dual of [(16318, 2), 32631, 4]-NRT-code), using
- linear OOA(8119, 177148, F81, 3, 7) (dual of [(177148, 3), 531425, 8]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8119, 531444, F81, 7) (dual of [531444, 531425, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(8119, 531441, F81, 7) (dual of [531441, 531422, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- 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(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(6) ⊂ Ce(5) [i] based on
- OOA 3-folding [i] based on linear OA(8119, 531444, F81, 7) (dual of [531444, 531425, 8]-code), using
- linear OOA(815, 16318, F81, 3, 3) (dual of [(16318, 3), 48949, 4]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(8124, 193466, F81, 3, 7) (dual of [(193466, 3), 580374, 8]-NRT-code), using
- net defined by OOA [i] based on linear OOA(8124, 193465, F81, 9, 7) (dual of [(193465, 9), 1741161, 8]-NRT-code), using
- digital (42, 57, 1198371)-net over F81, using
- net defined by OOA [i] based on linear OOA(8157, 1198371, F81, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(8157, 8388598, F81, 15) (dual of [8388598, 8388541, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(8157, large, F81, 15) (dual of [large, large−57, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8157, large, F81, 15) (dual of [large, large−57, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(8157, 8388598, F81, 15) (dual of [8388598, 8388541, 16]-code), using
- net defined by OOA [i] based on linear OOA(8157, 1198371, F81, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- digital (17, 24, 193465)-net over F81, using
(81−15, 81, large)-Net over F81 — Digital
Digital (66, 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
(81−15, 81, large)-Net in Base 81 — Upper bound on s
There is no (66, 81, large)-net in base 81, because
- 13 times m-reduction [i] would yield (66, 68, large)-net in base 81, but