Best Known (44−12, 44, s)-Nets in Base 9
(44−12, 44, 1095)-Net over F9 — Constructive and digital
Digital (32, 44, 1095)-net over F9, using
- 92 times duplication [i] based on digital (30, 42, 1095)-net over F9, using
- net defined by OOA [i] based on linear OOA(942, 1095, F9, 12, 12) (dual of [(1095, 12), 13098, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(942, 6570, F9, 12) (dual of [6570, 6528, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- 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(933, 6561, F9, 10) (dual of [6561, 6528, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(91, 9, F9, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- OA 6-folding and stacking [i] based on linear OA(942, 6570, F9, 12) (dual of [6570, 6528, 13]-code), using
- net defined by OOA [i] based on linear OOA(942, 1095, F9, 12, 12) (dual of [(1095, 12), 13098, 13]-NRT-code), using
(44−12, 44, 6573)-Net over F9 — Digital
Digital (32, 44, 6573)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(944, 6573, F9, 12) (dual of [6573, 6529, 13]-code), using
- construction XX applied to Ce(11) ⊂ Ce(9) ⊂ Ce(7) [i] based on
- 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(933, 6561, F9, 10) (dual of [6561, 6528, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(929, 6561, F9, 8) (dual of [6561, 6532, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(91, 10, F9, 1) (dual of [10, 9, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- linear OA(91, 2, F9, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(11) ⊂ Ce(9) ⊂ Ce(7) [i] based on
(44−12, 44, 3723146)-Net in Base 9 — Upper bound on s
There is no (32, 44, 3723147)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 969774 593807 394145 977823 570328 529375 359377 > 944 [i]