Best Known (49, 63, s)-Nets in Base 9
(49, 63, 8437)-Net over F9 — Constructive and digital
Digital (49, 63, 8437)-net over F9, using
- 91 times duplication [i] based on digital (48, 62, 8437)-net over F9, using
- net defined by OOA [i] based on linear OOA(962, 8437, F9, 14, 14) (dual of [(8437, 14), 118056, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(962, 59059, F9, 14) (dual of [59059, 58997, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(962, 59060, F9, 14) (dual of [59060, 58998, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(961, 59049, F9, 14) (dual of [59049, 58988, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(951, 59049, F9, 12) (dual of [59049, 58998, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(91, 11, F9, 1) (dual of [11, 10, 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(962, 59060, F9, 14) (dual of [59060, 58998, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(962, 59059, F9, 14) (dual of [59059, 58997, 15]-code), using
- net defined by OOA [i] based on linear OOA(962, 8437, F9, 14, 14) (dual of [(8437, 14), 118056, 15]-NRT-code), using
(49, 63, 56296)-Net over F9 — Digital
Digital (49, 63, 56296)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(963, 56296, F9, 14) (dual of [56296, 56233, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(963, 59062, F9, 14) (dual of [59062, 58999, 15]-code), using
- construction XX applied to Ce(13) ⊂ Ce(11) ⊂ Ce(10) [i] based on
- linear OA(961, 59049, F9, 14) (dual of [59049, 58988, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(951, 59049, F9, 12) (dual of [59049, 58998, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(946, 59049, F9, 11) (dual of [59049, 59003, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(91, 12, F9, 1) (dual of [12, 11, 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(963, 59062, F9, 14) (dual of [59062, 58999, 15]-code), using
(49, 63, large)-Net in Base 9 — Upper bound on s
There is no (49, 63, large)-net in base 9, because
- 12 times m-reduction [i] would yield (49, 51, large)-net in base 9, but