Best Known (82−21, 82, s)-Nets in Base 8
(82−21, 82, 819)-Net over F8 — Constructive and digital
Digital (61, 82, 819)-net over F8, using
- net defined by OOA [i] based on linear OOA(882, 819, F8, 21, 21) (dual of [(819, 21), 17117, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(882, 8191, F8, 21) (dual of [8191, 8109, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(882, 8194, F8, 21) (dual of [8194, 8112, 22]-code), using
- trace code [i] based on linear OA(6441, 4097, F64, 21) (dual of [4097, 4056, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- trace code [i] based on linear OA(6441, 4097, F64, 21) (dual of [4097, 4056, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(882, 8194, F8, 21) (dual of [8194, 8112, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(882, 8191, F8, 21) (dual of [8191, 8109, 22]-code), using
(82−21, 82, 8008)-Net over F8 — Digital
Digital (61, 82, 8008)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(882, 8008, F8, 21) (dual of [8008, 7926, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(882, 8194, F8, 21) (dual of [8194, 8112, 22]-code), using
- trace code [i] based on linear OA(6441, 4097, F64, 21) (dual of [4097, 4056, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- trace code [i] based on linear OA(6441, 4097, F64, 21) (dual of [4097, 4056, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(882, 8194, F8, 21) (dual of [8194, 8112, 22]-code), using
(82−21, 82, large)-Net in Base 8 — Upper bound on s
There is no (61, 82, large)-net in base 8, because
- 19 times m-reduction [i] would yield (61, 63, large)-net in base 8, but