Best Known (45−13, 45, s)-Nets in Base 9
(45−13, 45, 1094)-Net over F9 — Constructive and digital
Digital (32, 45, 1094)-net over F9, using
- net defined by OOA [i] based on linear OOA(945, 1094, F9, 13, 13) (dual of [(1094, 13), 14177, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(945, 6565, F9, 13) (dual of [6565, 6520, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(945, 6561, F9, 13) (dual of [6561, 6516, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(941, 6561, F9, 12) (dual of [6561, 6520, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(12) ⊂ Ce(11) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(945, 6565, F9, 13) (dual of [6565, 6520, 14]-code), using
(45−13, 45, 4020)-Net over F9 — Digital
Digital (32, 45, 4020)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(945, 4020, F9, 13) (dual of [4020, 3975, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(945, 6561, F9, 13) (dual of [6561, 6516, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(945, 6561, F9, 13) (dual of [6561, 6516, 14]-code), using
(45−13, 45, 3723146)-Net in Base 9 — Upper bound on s
There is no (32, 45, 3723147)-net in base 9, because
- 1 times m-reduction [i] would yield (32, 44, 3723147)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 969774 593807 394145 977823 570328 529375 359377 > 944 [i]