Best Known (41, 48, s)-Nets in Base 9
(41, 48, 1594537)-Net over F9 — Constructive and digital
Digital (41, 48, 1594537)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (2, 5, 212)-net over F9, using
- net defined by OOA [i] based on linear OOA(95, 212, F9, 3, 3) (dual of [(212, 3), 631, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(95, 212, F9, 2, 3) (dual of [(212, 2), 419, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(95, 212, F9, 3, 3) (dual of [(212, 3), 631, 4]-NRT-code), using
- digital (36, 43, 1594325)-net over F9, using
- net defined by OOA [i] based on linear OOA(943, 1594325, F9, 7, 7) (dual of [(1594325, 7), 11160232, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(943, 4782976, F9, 7) (dual of [4782976, 4782933, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(943, 4782969, F9, 7) (dual of [4782969, 4782926, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(936, 4782969, F9, 6) (dual of [4782969, 4782933, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(90, 7, F9, 0) (dual of [7, 7, 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(6) ⊂ Ce(5) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(943, 4782976, F9, 7) (dual of [4782976, 4782933, 8]-code), using
- net defined by OOA [i] based on linear OOA(943, 1594325, F9, 7, 7) (dual of [(1594325, 7), 11160232, 8]-NRT-code), using
- digital (2, 5, 212)-net over F9, using
(41, 48, 2796200)-Net in Base 9 — Constructive
(41, 48, 2796200)-net in base 9, using
- base change [i] based on digital (25, 32, 2796200)-net over F27, using
- 271 times duplication [i] based on digital (24, 31, 2796200)-net over F27, using
- net defined by OOA [i] based on linear OOA(2731, 2796200, F27, 7, 7) (dual of [(2796200, 7), 19573369, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2731, 8388601, F27, 7) (dual of [8388601, 8388570, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(2731, large, F27, 7) (dual of [large, large−31, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2731, large, F27, 7) (dual of [large, large−31, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2731, 8388601, F27, 7) (dual of [8388601, 8388570, 8]-code), using
- net defined by OOA [i] based on linear OOA(2731, 2796200, F27, 7, 7) (dual of [(2796200, 7), 19573369, 8]-NRT-code), using
- 271 times duplication [i] based on digital (24, 31, 2796200)-net over F27, using
(41, 48, large)-Net over F9 — Digital
Digital (41, 48, large)-net over F9, using
- 91 times duplication [i] based on digital (40, 47, large)-net over F9, using
(41, 48, large)-Net in Base 9 — Upper bound on s
There is no (41, 48, large)-net in base 9, because
- 5 times m-reduction [i] would yield (41, 43, large)-net in base 9, but