Best Known (70, 70+19, s)-Nets in Base 9
(70, 70+19, 6564)-Net over F9 — Constructive and digital
Digital (70, 89, 6564)-net over F9, using
- 91 times duplication [i] based on digital (69, 88, 6564)-net over F9, using
- net defined by OOA [i] based on linear OOA(988, 6564, F9, 19, 19) (dual of [(6564, 19), 124628, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(988, 59077, F9, 19) (dual of [59077, 58989, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- linear OA(981, 59050, F9, 19) (dual of [59050, 58969, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 910−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(961, 59050, F9, 13) (dual of [59050, 58989, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 910−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(97, 27, F9, 5) (dual of [27, 20, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(97, 73, F9, 5) (dual of [73, 66, 6]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- OOA 9-folding and stacking with additional row [i] based on linear OA(988, 59077, F9, 19) (dual of [59077, 58989, 20]-code), using
- net defined by OOA [i] based on linear OOA(988, 6564, F9, 19, 19) (dual of [(6564, 19), 124628, 20]-NRT-code), using
(70, 70+19, 59083)-Net over F9 — Digital
Digital (70, 89, 59083)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(989, 59083, F9, 19) (dual of [59083, 58994, 20]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(988, 59081, F9, 19) (dual of [59081, 58993, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(12) [i] based on
- linear OA(981, 59049, F9, 19) (dual of [59049, 58968, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(956, 59049, F9, 13) (dual of [59049, 58993, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(97, 32, F9, 5) (dual of [32, 25, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(97, 73, F9, 5) (dual of [73, 66, 6]-code), using
- construction X applied to Ce(18) ⊂ Ce(12) [i] based on
- linear OA(988, 59082, F9, 18) (dual of [59082, 58994, 19]-code), using Gilbert–Varšamov bound and bm = 988 > Vbs−1(k−1) = 8224 136702 443124 395013 884791 154226 847088 382657 031931 169574 707289 403894 849757 896521 [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(988, 59081, F9, 19) (dual of [59081, 58993, 20]-code), using
- construction X with Varšamov bound [i] based on
(70, 70+19, large)-Net in Base 9 — Upper bound on s
There is no (70, 89, large)-net in base 9, because
- 17 times m-reduction [i] would yield (70, 72, large)-net in base 9, but