Best Known (82, 82+19, s)-Nets in Base 9
(82, 82+19, 59051)-Net over F9 — Constructive and digital
Digital (82, 101, 59051)-net over F9, using
- net defined by OOA [i] based on linear OOA(9101, 59051, F9, 19, 19) (dual of [(59051, 19), 1121868, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(9101, 531460, F9, 19) (dual of [531460, 531359, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(9101, 531463, F9, 19) (dual of [531463, 531362, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(997, 531441, F9, 19) (dual of [531441, 531344, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(979, 531441, F9, 15) (dual of [531441, 531362, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(94, 22, F9, 3) (dual of [22, 18, 4]-code or 22-cap in PG(3,9)), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(9101, 531463, F9, 19) (dual of [531463, 531362, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(9101, 531460, F9, 19) (dual of [531460, 531359, 20]-code), using
(82, 82+19, 368159)-Net over F9 — Digital
Digital (82, 101, 368159)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9101, 368159, F9, 19) (dual of [368159, 368058, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(9101, 531458, F9, 19) (dual of [531458, 531357, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- linear OA(997, 531442, F9, 19) (dual of [531442, 531345, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 912−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(985, 531442, F9, 15) (dual of [531442, 531357, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 912−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(94, 16, F9, 3) (dual of [16, 12, 4]-code or 16-cap in PG(3,9)), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(9101, 531458, F9, 19) (dual of [531458, 531357, 20]-code), using
(82, 82+19, large)-Net in Base 9 — Upper bound on s
There is no (82, 101, large)-net in base 9, because
- 17 times m-reduction [i] would yield (82, 84, large)-net in base 9, but