Best Known (55, 55+19, s)-Nets in Base 8
(55, 55+19, 910)-Net over F8 — Constructive and digital
Digital (55, 74, 910)-net over F8, using
- net defined by OOA [i] based on linear OOA(874, 910, F8, 19, 19) (dual of [(910, 19), 17216, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(874, 8191, F8, 19) (dual of [8191, 8117, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(874, 8194, F8, 19) (dual of [8194, 8120, 20]-code), using
- trace code [i] based on linear OA(6437, 4097, F64, 19) (dual of [4097, 4060, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- trace code [i] based on linear OA(6437, 4097, F64, 19) (dual of [4097, 4060, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(874, 8194, F8, 19) (dual of [8194, 8120, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(874, 8191, F8, 19) (dual of [8191, 8117, 20]-code), using
(55, 55+19, 7732)-Net over F8 — Digital
Digital (55, 74, 7732)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(874, 7732, F8, 19) (dual of [7732, 7658, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(874, 8194, F8, 19) (dual of [8194, 8120, 20]-code), using
- trace code [i] based on linear OA(6437, 4097, F64, 19) (dual of [4097, 4060, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- trace code [i] based on linear OA(6437, 4097, F64, 19) (dual of [4097, 4060, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(874, 8194, F8, 19) (dual of [8194, 8120, 20]-code), using
(55, 55+19, large)-Net in Base 8 — Upper bound on s
There is no (55, 74, large)-net in base 8, because
- 17 times m-reduction [i] would yield (55, 57, large)-net in base 8, but