Best Known (63−17, 63, s)-Nets in Base 9
(63−17, 63, 821)-Net over F9 — Constructive and digital
Digital (46, 63, 821)-net over F9, using
- 91 times duplication [i] based on digital (45, 62, 821)-net over F9, using
- net defined by OOA [i] based on linear OOA(962, 821, F9, 17, 17) (dual of [(821, 17), 13895, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(962, 6569, F9, 17) (dual of [6569, 6507, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(962, 6570, F9, 17) (dual of [6570, 6508, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(961, 6561, F9, 17) (dual of [6561, 6500, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(953, 6561, F9, 15) (dual of [6561, 6508, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(16) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(962, 6570, F9, 17) (dual of [6570, 6508, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(962, 6569, F9, 17) (dual of [6569, 6507, 18]-code), using
- net defined by OOA [i] based on linear OOA(962, 821, F9, 17, 17) (dual of [(821, 17), 13895, 18]-NRT-code), using
(63−17, 63, 6572)-Net over F9 — Digital
Digital (46, 63, 6572)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(963, 6572, F9, 17) (dual of [6572, 6509, 18]-code), using
- construction XX applied to Ce(16) ⊂ Ce(14) ⊂ Ce(13) [i] based on
- linear OA(961, 6561, F9, 17) (dual of [6561, 6500, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(953, 6561, F9, 15) (dual of [6561, 6508, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- 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(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(16) ⊂ Ce(14) ⊂ Ce(13) [i] based on
(63−17, 63, large)-Net in Base 9 — Upper bound on s
There is no (46, 63, large)-net in base 9, because
- 15 times m-reduction [i] would yield (46, 48, large)-net in base 9, but