Best Known (94−15, 94, s)-Nets in Base 9
(94−15, 94, 683283)-Net over F9 — Constructive and digital
Digital (79, 94, 683283)-net over F9, using
- 91 times duplication [i] based on digital (78, 93, 683283)-net over F9, using
- net defined by OOA [i] based on linear OOA(993, 683283, F9, 15, 15) (dual of [(683283, 15), 10249152, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(993, 4782982, F9, 15) (dual of [4782982, 4782889, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(993, 4782984, F9, 15) (dual of [4782984, 4782891, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(992, 4782969, F9, 15) (dual of [4782969, 4782877, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(978, 4782969, F9, 13) (dual of [4782969, 4782891, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(91, 15, F9, 1) (dual of [15, 14, 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(14) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(993, 4782984, F9, 15) (dual of [4782984, 4782891, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(993, 4782982, F9, 15) (dual of [4782982, 4782889, 16]-code), using
- net defined by OOA [i] based on linear OOA(993, 683283, F9, 15, 15) (dual of [(683283, 15), 10249152, 16]-NRT-code), using
(94−15, 94, 4751379)-Net over F9 — Digital
Digital (79, 94, 4751379)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(994, 4751379, F9, 15) (dual of [4751379, 4751285, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(994, 4782986, F9, 15) (dual of [4782986, 4782892, 16]-code), using
- 1 times code embedding in larger space [i] based on linear OA(993, 4782985, F9, 15) (dual of [4782985, 4782892, 16]-code), using
- construction X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(992, 4782969, F9, 15) (dual of [4782969, 4782877, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(978, 4782969, F9, 13) (dual of [4782969, 4782891, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(915, 16, F9, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,9)), using
- dual of repetition code with length 16 [i]
- linear OA(91, 16, F9, 1) (dual of [16, 15, 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
- construction X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(993, 4782985, F9, 15) (dual of [4782985, 4782892, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(994, 4782986, F9, 15) (dual of [4782986, 4782892, 16]-code), using
(94−15, 94, large)-Net in Base 9 — Upper bound on s
There is no (79, 94, large)-net in base 9, because
- 13 times m-reduction [i] would yield (79, 81, large)-net in base 9, but