Best Known (21−6, 21, s)-Nets in Base 9
(21−6, 21, 2188)-Net over F9 — Constructive and digital
Digital (15, 21, 2188)-net over F9, using
- net defined by OOA [i] based on linear OOA(921, 2188, F9, 6, 6) (dual of [(2188, 6), 13107, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(921, 6564, F9, 6) (dual of [6564, 6543, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(921, 6565, F9, 6) (dual of [6565, 6544, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(921, 6561, F9, 6) (dual of [6561, 6540, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(917, 6561, F9, 5) (dual of [6561, 6544, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(90, 4, F9, 0) (dual of [4, 4, 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
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(921, 6565, F9, 6) (dual of [6565, 6544, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(921, 6564, F9, 6) (dual of [6564, 6543, 7]-code), using
(21−6, 21, 6565)-Net over F9 — Digital
Digital (15, 21, 6565)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(921, 6565, F9, 6) (dual of [6565, 6544, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(921, 6561, F9, 6) (dual of [6561, 6540, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(917, 6561, F9, 5) (dual of [6561, 6544, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(90, 4, F9, 0) (dual of [4, 4, 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
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
(21−6, 21, 1086402)-Net in Base 9 — Upper bound on s
There is no (15, 21, 1086403)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 109 419122 652289 862345 > 921 [i]