Best Known (178, 178+51, s)-Nets in Base 4
(178, 178+51, 1052)-Net over F4 — Constructive and digital
Digital (178, 229, 1052)-net over F4, using
- 41 times duplication [i] based on digital (177, 228, 1052)-net over F4, using
- trace code for nets [i] based on digital (6, 57, 263)-net over F256, using
- net from sequence [i] based on digital (6, 262)-sequence over F256, using
- trace code for nets [i] based on digital (6, 57, 263)-net over F256, using
(178, 178+51, 3994)-Net over F4 — Digital
Digital (178, 229, 3994)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4229, 3994, F4, 51) (dual of [3994, 3765, 52]-code), using
- discarding factors / shortening the dual code based on linear OA(4229, 4096, F4, 51) (dual of [4096, 3867, 52]-code), using
- an extension Ce(50) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,50], and designed minimum distance d ≥ |I|+1 = 51 [i]
- discarding factors / shortening the dual code based on linear OA(4229, 4096, F4, 51) (dual of [4096, 3867, 52]-code), using
(178, 178+51, 1050227)-Net in Base 4 — Upper bound on s
There is no (178, 229, 1050228)-net in base 4, because
- 1 times m-reduction [i] would yield (178, 228, 1050228)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 186074 008285 107009 242616 383684 100540 122536 960136 904510 068976 036093 539115 731405 104217 258204 148804 643751 763062 944188 974292 805831 218601 180036 > 4228 [i]