Best Known (82−21, 82, s)-Nets in Base 9
(82−21, 82, 1312)-Net over F9 — Constructive and digital
Digital (61, 82, 1312)-net over F9, using
- net defined by OOA [i] based on linear OOA(982, 1312, F9, 21, 21) (dual of [(1312, 21), 27470, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(982, 13121, F9, 21) (dual of [13121, 13039, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(982, 13124, F9, 21) (dual of [13124, 13042, 22]-code), using
- trace code [i] based on linear OA(8141, 6562, F81, 21) (dual of [6562, 6521, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- trace code [i] based on linear OA(8141, 6562, F81, 21) (dual of [6562, 6521, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(982, 13124, F9, 21) (dual of [13124, 13042, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(982, 13121, F9, 21) (dual of [13121, 13039, 22]-code), using
(82−21, 82, 11583)-Net over F9 — Digital
Digital (61, 82, 11583)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(982, 11583, F9, 21) (dual of [11583, 11501, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(982, 13124, F9, 21) (dual of [13124, 13042, 22]-code), using
- trace code [i] based on linear OA(8141, 6562, F81, 21) (dual of [6562, 6521, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- trace code [i] based on linear OA(8141, 6562, F81, 21) (dual of [6562, 6521, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(982, 13124, F9, 21) (dual of [13124, 13042, 22]-code), using
(82−21, 82, large)-Net in Base 9 — Upper bound on s
There is no (61, 82, large)-net in base 9, because
- 19 times m-reduction [i] would yield (61, 63, large)-net in base 9, but