Best Known (64, 64+21, s)-Nets in Base 9
(64, 64+21, 1313)-Net over F9 — Constructive and digital
Digital (64, 85, 1313)-net over F9, using
- 91 times duplication [i] based on digital (63, 84, 1313)-net over F9, using
- net defined by OOA [i] based on linear OOA(984, 1313, F9, 21, 21) (dual of [(1313, 21), 27489, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(984, 13131, F9, 21) (dual of [13131, 13047, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(984, 13134, F9, 21) (dual of [13134, 13050, 22]-code), using
- trace code [i] based on linear OA(8142, 6567, F81, 21) (dual of [6567, 6525, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(8141, 6562, F81, 21) (dual of [6562, 6521, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- 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(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([0,10]) ⊂ C([0,9]) [i] based on
- trace code [i] based on linear OA(8142, 6567, F81, 21) (dual of [6567, 6525, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(984, 13134, F9, 21) (dual of [13134, 13050, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(984, 13131, F9, 21) (dual of [13131, 13047, 22]-code), using
- net defined by OOA [i] based on linear OOA(984, 1313, F9, 21, 21) (dual of [(1313, 21), 27489, 22]-NRT-code), using
(64, 64+21, 13136)-Net over F9 — Digital
Digital (64, 85, 13136)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(985, 13136, F9, 21) (dual of [13136, 13051, 22]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(984, 13134, F9, 21) (dual of [13134, 13050, 22]-code), using
- trace code [i] based on linear OA(8142, 6567, F81, 21) (dual of [6567, 6525, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(8141, 6562, F81, 21) (dual of [6562, 6521, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- 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(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([0,10]) ⊂ C([0,9]) [i] based on
- trace code [i] based on linear OA(8142, 6567, F81, 21) (dual of [6567, 6525, 22]-code), using
- linear OA(984, 13135, F9, 20) (dual of [13135, 13051, 21]-code), using Gilbert–Varšamov bound and bm = 984 > Vbs−1(k−1) = 2 077686 364189 490985 883127 648593 333937 627420 666077 625744 043499 089149 751017 181489 [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(984, 13134, F9, 21) (dual of [13134, 13050, 22]-code), using
- construction X with Varšamov bound [i] based on
(64, 64+21, large)-Net in Base 9 — Upper bound on s
There is no (64, 85, large)-net in base 9, because
- 19 times m-reduction [i] would yield (64, 66, large)-net in base 9, but