Best Known (21−5, 21, s)-Nets in Base 9
(21−5, 21, 29526)-Net over F9 — Constructive and digital
Digital (16, 21, 29526)-net over F9, using
- net defined by OOA [i] based on linear OOA(921, 29526, F9, 5, 5) (dual of [(29526, 5), 147609, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(921, 59053, F9, 5) (dual of [59053, 59032, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(921, 59054, F9, 5) (dual of [59054, 59033, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(921, 59049, F9, 5) (dual of [59049, 59028, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(916, 59049, F9, 4) (dual of [59049, 59033, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(90, 5, F9, 0) (dual of [5, 5, 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(4) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(921, 59054, F9, 5) (dual of [59054, 59033, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(921, 59053, F9, 5) (dual of [59053, 59032, 6]-code), using
(21−5, 21, 59054)-Net over F9 — Digital
Digital (16, 21, 59054)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(921, 59054, F9, 5) (dual of [59054, 59033, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(921, 59049, F9, 5) (dual of [59049, 59028, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(916, 59049, F9, 4) (dual of [59049, 59033, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(90, 5, F9, 0) (dual of [5, 5, 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(4) ⊂ Ce(3) [i] based on
(21−5, 21, large)-Net in Base 9 — Upper bound on s
There is no (16, 21, large)-net in base 9, because
- 3 times m-reduction [i] would yield (16, 18, large)-net in base 9, but