Best Known (48−13, 48, s)-Nets in Base 9
(48−13, 48, 1095)-Net over F9 — Constructive and digital
Digital (35, 48, 1095)-net over F9, using
- 91 times duplication [i] based on digital (34, 47, 1095)-net over F9, using
- net defined by OOA [i] based on linear OOA(947, 1095, F9, 13, 13) (dual of [(1095, 13), 14188, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(947, 6571, F9, 13) (dual of [6571, 6524, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(947, 6572, F9, 13) (dual of [6572, 6525, 14]-code), using
- construction XX applied to Ce(12) ⊂ Ce(10) ⊂ Ce(9) [i] based on
- linear OA(945, 6561, F9, 13) (dual of [6561, 6516, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(937, 6561, F9, 11) (dual of [6561, 6524, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(933, 6561, F9, 10) (dual of [6561, 6528, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(91, 10, F9, 1) (dual of [10, 9, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- linear OA(90, 1, F9, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(12) ⊂ Ce(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(947, 6572, F9, 13) (dual of [6572, 6525, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(947, 6571, F9, 13) (dual of [6571, 6524, 14]-code), using
- net defined by OOA [i] based on linear OOA(947, 1095, F9, 13, 13) (dual of [(1095, 13), 14188, 14]-NRT-code), using
(48−13, 48, 6576)-Net over F9 — Digital
Digital (35, 48, 6576)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(948, 6576, F9, 13) (dual of [6576, 6528, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(945, 6561, F9, 13) (dual of [6561, 6516, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(933, 6561, F9, 10) (dual of [6561, 6528, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(93, 15, F9, 2) (dual of [15, 12, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 92−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
(48−13, 48, large)-Net in Base 9 — Upper bound on s
There is no (35, 48, large)-net in base 9, because
- 11 times m-reduction [i] would yield (35, 37, large)-net in base 9, but