Best Known (58−29, 58, s)-Nets in Base 81
(58−29, 58, 469)-Net over F81 — Constructive and digital
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
(58−29, 58, 1998)-Net over F81 — Digital
Digital (29, 58, 1998)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8158, 1998, F81, 3, 29) (dual of [(1998, 3), 5936, 30]-NRT-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OOA(8158, 2189, F81, 3, 29) (dual of [(2189, 3), 6509, 30]-NRT-code), using
(58−29, 58, 4452742)-Net in Base 81 — Upper bound on s
There is no (29, 58, 4452743)-net in base 81, because
- 1 times m-reduction [i] would yield (29, 57, 4452743)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 6 076414 542998 357533 837584 623069 580458 782477 400536 924465 650164 465581 480512 756502 299158 689629 359784 605402 022561 > 8157 [i]