Best Known (83−13, 83, s)-Nets in Base 9
(83−13, 83, 797165)-Net over F9 — Constructive and digital
Digital (70, 83, 797165)-net over F9, using
- 92 times duplication [i] based on digital (68, 81, 797165)-net over F9, using
- net defined by OOA [i] based on linear OOA(981, 797165, F9, 13, 13) (dual of [(797165, 13), 10363064, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(981, 4782991, F9, 13) (dual of [4782991, 4782910, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(981, 4782993, F9, 13) (dual of [4782993, 4782912, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(978, 4782969, F9, 13) (dual of [4782969, 4782891, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(957, 4782969, F9, 10) (dual of [4782969, 4782912, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(93, 24, F9, 2) (dual of [24, 21, 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
- discarding factors / shortening the dual code based on linear OA(981, 4782993, F9, 13) (dual of [4782993, 4782912, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(981, 4782991, F9, 13) (dual of [4782991, 4782910, 14]-code), using
- net defined by OOA [i] based on linear OOA(981, 797165, F9, 13, 13) (dual of [(797165, 13), 10363064, 14]-NRT-code), using
(83−13, 83, 4782997)-Net over F9 — Digital
Digital (70, 83, 4782997)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(983, 4782997, F9, 13) (dual of [4782997, 4782914, 14]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(981, 4782993, F9, 13) (dual of [4782993, 4782912, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(978, 4782969, F9, 13) (dual of [4782969, 4782891, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(957, 4782969, F9, 10) (dual of [4782969, 4782912, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(93, 24, F9, 2) (dual of [24, 21, 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
- linear OA(981, 4782995, F9, 12) (dual of [4782995, 4782914, 13]-code), using Gilbert–Varšamov bound and bm = 981 > Vbs−1(k−1) = 6449 521079 776401 215371 956098 495556 207324 773567 379523 843934 556889 342143 427281 [i]
- linear OA(90, 2, F9, 0) (dual of [2, 2, 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(981, 4782993, F9, 13) (dual of [4782993, 4782912, 14]-code), using
- construction X with Varšamov bound [i] based on
(83−13, 83, large)-Net in Base 9 — Upper bound on s
There is no (70, 83, large)-net in base 9, because
- 11 times m-reduction [i] would yield (70, 72, large)-net in base 9, but