Best Known (10, 10+4, s)-Nets in Base 9
(10, 10+4, 6563)-Net over F9 — Constructive and digital
Digital (10, 14, 6563)-net over F9, using
- net defined by OOA [i] based on linear OOA(914, 6563, F9, 4, 4) (dual of [(6563, 4), 26238, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(914, 6563, F9, 3, 4) (dual of [(6563, 3), 19675, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(914, 13126, F9, 4) (dual of [13126, 13112, 5]-code), using
- trace code [i] based on linear OA(817, 6563, F81, 4) (dual of [6563, 6556, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(817, 6561, F81, 4) (dual of [6561, 6554, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(815, 6561, F81, 3) (dual of [6561, 6556, 4]-code or 6561-cap in PG(4,81)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [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(3) ⊂ Ce(2) [i] based on
- trace code [i] based on linear OA(817, 6563, F81, 4) (dual of [6563, 6556, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(914, 13126, F9, 4) (dual of [13126, 13112, 5]-code), using
- appending kth column [i] based on linear OOA(914, 6563, F9, 3, 4) (dual of [(6563, 3), 19675, 5]-NRT-code), using
(10, 10+4, 13126)-Net over F9 — Digital
Digital (10, 14, 13126)-net over F9, using
- net defined by OOA [i] based on linear OOA(914, 13126, F9, 4, 4) (dual of [(13126, 4), 52490, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(914, 13126, F9, 3, 4) (dual of [(13126, 3), 39364, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(914, 13126, F9, 4) (dual of [13126, 13112, 5]-code), using
- trace code [i] based on linear OA(817, 6563, F81, 4) (dual of [6563, 6556, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(817, 6561, F81, 4) (dual of [6561, 6554, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(815, 6561, F81, 3) (dual of [6561, 6556, 4]-code or 6561-cap in PG(4,81)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [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(3) ⊂ Ce(2) [i] based on
- trace code [i] based on linear OA(817, 6563, F81, 4) (dual of [6563, 6556, 5]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(914, 13126, F9, 4) (dual of [13126, 13112, 5]-code), using
- appending kth column [i] based on linear OOA(914, 13126, F9, 3, 4) (dual of [(13126, 3), 39364, 5]-NRT-code), using
(10, 10+4, 845516)-Net in Base 9 — Upper bound on s
There is no (10, 14, 845517)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 22 876808 498065 > 914 [i]