Best Known (23, 31, s)-Nets in Base 9
(23, 31, 3281)-Net over F9 — Constructive and digital
Digital (23, 31, 3281)-net over F9, using
- 91 times duplication [i] based on digital (22, 30, 3281)-net over F9, using
- net defined by OOA [i] based on linear OOA(930, 3281, F9, 8, 8) (dual of [(3281, 8), 26218, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(930, 13124, F9, 8) (dual of [13124, 13094, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(930, 13126, F9, 8) (dual of [13126, 13096, 9]-code), using
- trace code [i] based on linear OA(8115, 6563, F81, 8) (dual of [6563, 6548, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(8115, 6561, F81, 8) (dual of [6561, 6546, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(8113, 6561, F81, 7) (dual of [6561, 6548, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(7) ⊂ Ce(6) [i] based on
- trace code [i] based on linear OA(8115, 6563, F81, 8) (dual of [6563, 6548, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(930, 13126, F9, 8) (dual of [13126, 13096, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(930, 13124, F9, 8) (dual of [13124, 13094, 9]-code), using
- net defined by OOA [i] based on linear OOA(930, 3281, F9, 8, 8) (dual of [(3281, 8), 26218, 9]-NRT-code), using
(23, 31, 13128)-Net over F9 — Digital
Digital (23, 31, 13128)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(931, 13128, F9, 8) (dual of [13128, 13097, 9]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(930, 13126, F9, 8) (dual of [13126, 13096, 9]-code), using
- trace code [i] based on linear OA(8115, 6563, F81, 8) (dual of [6563, 6548, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(8115, 6561, F81, 8) (dual of [6561, 6546, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(8113, 6561, F81, 7) (dual of [6561, 6548, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(7) ⊂ Ce(6) [i] based on
- trace code [i] based on linear OA(8115, 6563, F81, 8) (dual of [6563, 6548, 9]-code), using
- linear OA(930, 13127, F9, 7) (dual of [13127, 13097, 8]-code), using Gilbert–Varšamov bound and bm = 930 > Vbs−1(k−1) = 1860 074255 197268 143189 569009 [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(930, 13126, F9, 8) (dual of [13126, 13096, 9]-code), using
- construction X with Varšamov bound [i] based on
(23, 31, 6876099)-Net in Base 9 — Upper bound on s
There is no (23, 31, 6876100)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 381520 560783 387407 094010 390401 > 931 [i]