Best Known (77−19, 77, s)-Nets in Base 9
(77−19, 77, 1459)-Net over F9 — Constructive and digital
Digital (58, 77, 1459)-net over F9, using
- 91 times duplication [i] based on digital (57, 76, 1459)-net over F9, using
- net defined by OOA [i] based on linear OOA(976, 1459, F9, 19, 19) (dual of [(1459, 19), 27645, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(976, 13132, F9, 19) (dual of [13132, 13056, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(976, 13134, F9, 19) (dual of [13134, 13058, 20]-code), using
- trace code [i] based on linear OA(8138, 6567, F81, 19) (dual of [6567, 6529, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(8137, 6562, F81, 19) (dual of [6562, 6525, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(8133, 6562, F81, 17) (dual of [6562, 6529, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (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([0,9]) ⊂ C([0,8]) [i] based on
- trace code [i] based on linear OA(8138, 6567, F81, 19) (dual of [6567, 6529, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(976, 13134, F9, 19) (dual of [13134, 13058, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(976, 13132, F9, 19) (dual of [13132, 13056, 20]-code), using
- net defined by OOA [i] based on linear OOA(976, 1459, F9, 19, 19) (dual of [(1459, 19), 27645, 20]-NRT-code), using
(77−19, 77, 13136)-Net over F9 — Digital
Digital (58, 77, 13136)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(977, 13136, F9, 19) (dual of [13136, 13059, 20]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(976, 13134, F9, 19) (dual of [13134, 13058, 20]-code), using
- trace code [i] based on linear OA(8138, 6567, F81, 19) (dual of [6567, 6529, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(8137, 6562, F81, 19) (dual of [6562, 6525, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(8133, 6562, F81, 17) (dual of [6562, 6529, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (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([0,9]) ⊂ C([0,8]) [i] based on
- trace code [i] based on linear OA(8138, 6567, F81, 19) (dual of [6567, 6529, 20]-code), using
- linear OA(976, 13135, F9, 18) (dual of [13135, 13059, 19]-code), using Gilbert–Varšamov bound and bm = 976 > Vbs−1(k−1) = 64533 007435 981594 478124 168139 266388 015232 798430 043813 615160 566038 237489 [i]
- linear OA(90, 1, F9, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(976, 13134, F9, 19) (dual of [13134, 13058, 20]-code), using
- construction X with Varšamov bound [i] based on
(77−19, 77, large)-Net in Base 9 — Upper bound on s
There is no (58, 77, large)-net in base 9, because
- 17 times m-reduction [i] would yield (58, 60, large)-net in base 9, but