Best Known (59−29, 59, s)-Nets in Base 81
(59−29, 59, 469)-Net over F81 — Constructive and digital
Digital (30, 59, 469)-net over F81, using
- 811 times duplication [i] based on digital (29, 58, 469)-net over F81, using
- net defined by OOA [i] based on linear OOA(8158, 469, F81, 29, 29) (dual of [(469, 29), 13543, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(8158, 6567, F81, 29) (dual of [6567, 6509, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,13]) [i] based on
- linear OA(8157, 6562, F81, 29) (dual of [6562, 6505, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(8153, 6562, F81, 27) (dual of [6562, 6509, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 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([0,14]) ⊂ C([0,13]) [i] based on
- OOA 14-folding and stacking with additional row [i] based on linear OA(8158, 6567, F81, 29) (dual of [6567, 6509, 30]-code), using
- net defined by OOA [i] based on linear OOA(8158, 469, F81, 29, 29) (dual of [(469, 29), 13543, 30]-NRT-code), using
(59−29, 59, 2189)-Net over F81 — Digital
Digital (30, 59, 2189)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8159, 2189, F81, 3, 29) (dual of [(2189, 3), 6508, 30]-NRT-code), using
- 811 times duplication [i] based on linear OOA(8158, 2189, F81, 3, 29) (dual of [(2189, 3), 6509, 30]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8158, 6567, F81, 29) (dual of [6567, 6509, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,13]) [i] based on
- linear OA(8157, 6562, F81, 29) (dual of [6562, 6505, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(8153, 6562, F81, 27) (dual of [6562, 6509, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 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([0,14]) ⊂ C([0,13]) [i] based on
- OOA 3-folding [i] based on linear OA(8158, 6567, F81, 29) (dual of [6567, 6509, 30]-code), using
- 811 times duplication [i] based on linear OOA(8158, 2189, F81, 3, 29) (dual of [(2189, 3), 6509, 30]-NRT-code), using
(59−29, 59, 6094640)-Net in Base 81 — Upper bound on s
There is no (30, 59, 6094641)-net in base 81, because
- 1 times m-reduction [i] would yield (30, 58, 6094641)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 492 189030 192746 261768 076643 051619 055814 248052 872992 354854 871615 258686 139612 481614 790997 456818 752701 065255 252321 > 8158 [i]