Best Known (62−11, 62, s)-Nets in Base 81
(62−11, 62, 3362002)-Net over F81 — Constructive and digital
Digital (51, 62, 3362002)-net over F81, using
- generalized (u, u+v)-construction [i] based on
- digital (1, 4, 6562)-net over F81, using
- net defined by OOA [i] based on linear OOA(814, 6562, F81, 3, 3) (dual of [(6562, 3), 19682, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(814, 6562, F81, 2, 3) (dual of [(6562, 2), 13120, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(814, 6562, F81, 3, 3) (dual of [(6562, 3), 19682, 4]-NRT-code), using
- digital (12, 17, 1677720)-net over F81, using
- s-reduction based on digital (12, 17, 4194301)-net over F81, using
- net defined by OOA [i] based on linear OOA(8117, 4194301, F81, 5, 5) (dual of [(4194301, 5), 20971488, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(8117, 4194301, F81, 4, 5) (dual of [(4194301, 4), 16777187, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(8117, large, F81, 5) (dual of [large, large−17, 6]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- OOA 2-folding and stacking with additional row [i] based on linear OA(8117, large, F81, 5) (dual of [large, large−17, 6]-code), using
- appending kth column [i] based on linear OOA(8117, 4194301, F81, 4, 5) (dual of [(4194301, 4), 16777187, 6]-NRT-code), using
- net defined by OOA [i] based on linear OOA(8117, 4194301, F81, 5, 5) (dual of [(4194301, 5), 20971488, 6]-NRT-code), using
- s-reduction based on digital (12, 17, 4194301)-net over F81, using
- digital (30, 41, 1677720)-net over F81, using
- net defined by OOA [i] based on linear OOA(8141, 1677720, F81, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(8141, 8388601, F81, 11) (dual of [8388601, 8388560, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(8141, large, F81, 11) (dual of [large, large−41, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8141, large, F81, 11) (dual of [large, large−41, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(8141, 8388601, F81, 11) (dual of [8388601, 8388560, 12]-code), using
- net defined by OOA [i] based on linear OOA(8141, 1677720, F81, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- digital (1, 4, 6562)-net over F81, using
(62−11, 62, large)-Net over F81 — Digital
Digital (51, 62, large)-net over F81, using
- 7 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
(62−11, 62, large)-Net in Base 81 — Upper bound on s
There is no (51, 62, large)-net in base 81, because
- 9 times m-reduction [i] would yield (51, 53, large)-net in base 81, but