Best Known (51−14, 51, s)-Nets in Base 9
(51−14, 51, 938)-Net over F9 — Constructive and digital
Digital (37, 51, 938)-net over F9, using
- 91 times duplication [i] based on digital (36, 50, 938)-net over F9, using
- net defined by OOA [i] based on linear OOA(950, 938, F9, 14, 14) (dual of [(938, 14), 13082, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(950, 6566, F9, 14) (dual of [6566, 6516, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(950, 6570, F9, 14) (dual of [6570, 6520, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(949, 6561, F9, 14) (dual of [6561, 6512, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(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(13) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(950, 6570, F9, 14) (dual of [6570, 6520, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(950, 6566, F9, 14) (dual of [6566, 6516, 15]-code), using
- net defined by OOA [i] based on linear OOA(950, 938, F9, 14, 14) (dual of [(938, 14), 13082, 15]-NRT-code), using
(51−14, 51, 6249)-Net over F9 — Digital
Digital (37, 51, 6249)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(951, 6249, F9, 14) (dual of [6249, 6198, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(951, 6572, F9, 14) (dual of [6572, 6521, 15]-code), using
- construction XX applied to Ce(13) ⊂ Ce(11) ⊂ Ce(10) [i] based on
- linear OA(949, 6561, F9, 14) (dual of [6561, 6512, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(937, 6561, F9, 11) (dual of [6561, 6524, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(90, 1, F9, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(13) ⊂ Ce(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(951, 6572, F9, 14) (dual of [6572, 6521, 15]-code), using
(51−14, 51, 3785876)-Net in Base 9 — Upper bound on s
There is no (37, 51, 3785877)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 4 638397 714851 736498 736552 406574 762521 193515 896665 > 951 [i]