Best Known (99, 99+19, s)-Nets in Base 9
(99, 99+19, 531443)-Net over F9 — Constructive and digital
Digital (99, 118, 531443)-net over F9, using
- 91 times duplication [i] based on digital (98, 117, 531443)-net over F9, using
- net defined by OOA [i] based on linear OOA(9117, 531443, F9, 19, 19) (dual of [(531443, 19), 10097300, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(9117, 4782988, F9, 19) (dual of [4782988, 4782871, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- linear OA(9113, 4782970, F9, 19) (dual of [4782970, 4782857, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 914−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(999, 4782970, F9, 15) (dual of [4782970, 4782871, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 914−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(94, 18, F9, 3) (dual of [18, 14, 4]-code or 18-cap in PG(3,9)), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- OOA 9-folding and stacking with additional row [i] based on linear OA(9117, 4782988, F9, 19) (dual of [4782988, 4782871, 20]-code), using
- net defined by OOA [i] based on linear OOA(9117, 531443, F9, 19, 19) (dual of [(531443, 19), 10097300, 20]-NRT-code), using
(99, 99+19, 3313506)-Net over F9 — Digital
Digital (99, 118, 3313506)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9118, 3313506, F9, 19) (dual of [3313506, 3313388, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(9118, 4782996, F9, 19) (dual of [4782996, 4782878, 20]-code), using
- construction XX applied to Ce(18) ⊂ Ce(14) ⊂ Ce(13) [i] based on
- linear OA(9113, 4782969, F9, 19) (dual of [4782969, 4782856, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 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(985, 4782969, F9, 14) (dual of [4782969, 4782884, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(94, 26, F9, 3) (dual of [26, 22, 4]-code or 26-cap in PG(3,9)), 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(18) ⊂ Ce(14) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(9118, 4782996, F9, 19) (dual of [4782996, 4782878, 20]-code), using
(99, 99+19, large)-Net in Base 9 — Upper bound on s
There is no (99, 118, large)-net in base 9, because
- 17 times m-reduction [i] would yield (99, 101, large)-net in base 9, but