Best Known (60−13, 60, s)-Nets in Base 9
(60−13, 60, 9844)-Net over F9 — Constructive and digital
Digital (47, 60, 9844)-net over F9, using
- 91 times duplication [i] based on digital (46, 59, 9844)-net over F9, using
- net defined by OOA [i] based on linear OOA(959, 9844, F9, 13, 13) (dual of [(9844, 13), 127913, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(959, 59065, F9, 13) (dual of [59065, 59006, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(959, 59067, F9, 13) (dual of [59067, 59008, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- 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(941, 59049, F9, 10) (dual of [59049, 59008, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(93, 18, F9, 2) (dual of [18, 15, 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(959, 59067, F9, 13) (dual of [59067, 59008, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(959, 59065, F9, 13) (dual of [59065, 59006, 14]-code), using
- net defined by OOA [i] based on linear OOA(959, 9844, F9, 13, 13) (dual of [(9844, 13), 127913, 14]-NRT-code), using
(60−13, 60, 59069)-Net over F9 — Digital
Digital (47, 60, 59069)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(960, 59069, F9, 13) (dual of [59069, 59009, 14]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(959, 59067, F9, 13) (dual of [59067, 59008, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- 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(941, 59049, F9, 10) (dual of [59049, 59008, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(93, 18, F9, 2) (dual of [18, 15, 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(959, 59068, F9, 12) (dual of [59068, 59009, 13]-code), using Gilbert–Varšamov bound and bm = 959 > Vbs−1(k−1) = 6 564987 229295 981413 666846 976743 171770 549574 616591 158169 [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(959, 59067, F9, 13) (dual of [59067, 59008, 14]-code), using
- construction X with Varšamov bound [i] based on
(60−13, 60, large)-Net in Base 9 — Upper bound on s
There is no (47, 60, large)-net in base 9, because
- 11 times m-reduction [i] would yield (47, 49, large)-net in base 9, but