Best Known (23−6, 23, s)-Nets in Base 9
(23−6, 23, 4375)-Net over F9 — Constructive and digital
Digital (17, 23, 4375)-net over F9, using
- 91 times duplication [i] based on digital (16, 22, 4375)-net over F9, using
- net defined by OOA [i] based on linear OOA(922, 4375, F9, 6, 6) (dual of [(4375, 6), 26228, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(922, 13125, F9, 6) (dual of [13125, 13103, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(922, 13126, F9, 6) (dual of [13126, 13104, 7]-code), using
- trace code [i] based on linear OA(8111, 6563, F81, 6) (dual of [6563, 6552, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(8111, 6561, F81, 6) (dual of [6561, 6550, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(819, 6561, F81, 5) (dual of [6561, 6552, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(810, 2, F81, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- trace code [i] based on linear OA(8111, 6563, F81, 6) (dual of [6563, 6552, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(922, 13126, F9, 6) (dual of [13126, 13104, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(922, 13125, F9, 6) (dual of [13125, 13103, 7]-code), using
- net defined by OOA [i] based on linear OOA(922, 4375, F9, 6, 6) (dual of [(4375, 6), 26228, 7]-NRT-code), using
(23−6, 23, 13128)-Net over F9 — Digital
Digital (17, 23, 13128)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(923, 13128, F9, 6) (dual of [13128, 13105, 7]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(922, 13126, F9, 6) (dual of [13126, 13104, 7]-code), using
- trace code [i] based on linear OA(8111, 6563, F81, 6) (dual of [6563, 6552, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(8111, 6561, F81, 6) (dual of [6561, 6550, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(819, 6561, F81, 5) (dual of [6561, 6552, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(810, 2, F81, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- trace code [i] based on linear OA(8111, 6563, F81, 6) (dual of [6563, 6552, 7]-code), using
- linear OA(922, 13127, F9, 5) (dual of [13127, 13105, 6]-code), using Gilbert–Varšamov bound and bm = 922 > Vbs−1(k−1) = 5 064030 538616 385009 [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(922, 13126, F9, 6) (dual of [13126, 13104, 7]-code), using
- construction X with Varšamov bound [i] based on
(23−6, 23, 4700595)-Net in Base 9 — Upper bound on s
There is no (17, 23, 4700596)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 8862 941275 574159 821409 > 923 [i]