Best Known (55, 55+19, s)-Nets in Base 16
(55, 55+19, 14563)-Net over F16 — Constructive and digital
Digital (55, 74, 14563)-net over F16, using
- net defined by OOA [i] based on linear OOA(1674, 14563, F16, 19, 19) (dual of [(14563, 19), 276623, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(1674, 131068, F16, 19) (dual of [131068, 130994, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(1674, 131074, F16, 19) (dual of [131074, 131000, 20]-code), using
- trace code [i] based on linear OA(25637, 65537, F256, 19) (dual of [65537, 65500, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- trace code [i] based on linear OA(25637, 65537, F256, 19) (dual of [65537, 65500, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(1674, 131074, F16, 19) (dual of [131074, 131000, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(1674, 131068, F16, 19) (dual of [131068, 130994, 20]-code), using
(55, 55+19, 70865)-Net over F16 — Digital
Digital (55, 74, 70865)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1674, 70865, F16, 19) (dual of [70865, 70791, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(1674, 131074, F16, 19) (dual of [131074, 131000, 20]-code), using
- trace code [i] based on linear OA(25637, 65537, F256, 19) (dual of [65537, 65500, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- trace code [i] based on linear OA(25637, 65537, F256, 19) (dual of [65537, 65500, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(1674, 131074, F16, 19) (dual of [131074, 131000, 20]-code), using
(55, 55+19, large)-Net in Base 16 — Upper bound on s
There is no (55, 74, large)-net in base 16, because
- 17 times m-reduction [i] would yield (55, 57, large)-net in base 16, but