Best Known (216−48, 216, s)-Nets in Base 4
(216−48, 216, 1052)-Net over F4 — Constructive and digital
Digital (168, 216, 1052)-net over F4, using
- trace code for nets [i] based on digital (6, 54, 263)-net over F256, using
- net from sequence [i] based on digital (6, 262)-sequence over F256, using
(216−48, 216, 3873)-Net over F4 — Digital
Digital (168, 216, 3873)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4216, 3873, F4, 48) (dual of [3873, 3657, 49]-code), using
- discarding factors / shortening the dual code based on linear OA(4216, 4096, F4, 48) (dual of [4096, 3880, 49]-code), using
- 1 times truncation [i] based on linear OA(4217, 4097, F4, 49) (dual of [4097, 3880, 50]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,24], and minimum distance d ≥ |{−24,−23,…,24}|+1 = 50 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(4217, 4097, F4, 49) (dual of [4097, 3880, 50]-code), using
- discarding factors / shortening the dual code based on linear OA(4216, 4096, F4, 48) (dual of [4096, 3880, 49]-code), using
(216−48, 216, 856587)-Net in Base 4 — Upper bound on s
There is no (168, 216, 856588)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 11090 966198 657357 202468 004543 207115 035867 104911 397899 992481 577217 939018 926679 139453 123214 217325 192466 289552 423908 979453 958190 084256 > 4216 [i]