Best Known (56−19, 56, s)-Nets in Base 81
(56−19, 56, 59049)-Net over F81 — Constructive and digital
Digital (37, 56, 59049)-net over F81, using
- 811 times duplication [i] based on digital (36, 55, 59049)-net over F81, using
- net defined by OOA [i] based on linear OOA(8155, 59049, F81, 19, 19) (dual of [(59049, 19), 1121876, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(8155, 531442, F81, 19) (dual of [531442, 531387, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- OOA 9-folding and stacking with additional row [i] based on linear OA(8155, 531442, F81, 19) (dual of [531442, 531387, 20]-code), using
- net defined by OOA [i] based on linear OOA(8155, 59049, F81, 19, 19) (dual of [(59049, 19), 1121876, 20]-NRT-code), using
(56−19, 56, 234732)-Net over F81 — Digital
Digital (37, 56, 234732)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8156, 234732, F81, 2, 19) (dual of [(234732, 2), 469408, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8156, 265724, F81, 2, 19) (dual of [(265724, 2), 531392, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8156, 531448, F81, 19) (dual of [531448, 531392, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(8156, 531449, F81, 19) (dual of [531449, 531393, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(8155, 531442, F81, 19) (dual of [531442, 531387, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(8149, 531442, F81, 17) (dual of [531442, 531393, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(811, 7, F81, 1) (dual of [7, 6, 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 C([0,9]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8156, 531449, F81, 19) (dual of [531449, 531393, 20]-code), using
- OOA 2-folding [i] based on linear OA(8156, 531448, F81, 19) (dual of [531448, 531392, 20]-code), using
- discarding factors / shortening the dual code based on linear OOA(8156, 265724, F81, 2, 19) (dual of [(265724, 2), 531392, 20]-NRT-code), using
(56−19, 56, large)-Net in Base 81 — Upper bound on s
There is no (37, 56, large)-net in base 81, because
- 17 times m-reduction [i] would yield (37, 39, large)-net in base 81, but