Best Known (199, 199+57, s)-Nets in Base 4
(199, 199+57, 1539)-Net over F4 — Constructive and digital
Digital (199, 256, 1539)-net over F4, using
- 41 times duplication [i] based on digital (198, 255, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 85, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- trace code for nets [i] based on digital (28, 85, 513)-net over F64, using
(199, 199+57, 4160)-Net over F4 — Digital
Digital (199, 256, 4160)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4256, 4160, F4, 57) (dual of [4160, 3904, 58]-code), using
- 60 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0, 1, 14 times 0, 1, 40 times 0) [i] based on linear OA(4253, 4097, F4, 57) (dual of [4097, 3844, 58]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,28], and minimum distance d ≥ |{−28,−27,…,28}|+1 = 58 (BCH-bound) [i]
- 60 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0, 1, 14 times 0, 1, 40 times 0) [i] based on linear OA(4253, 4097, F4, 57) (dual of [4097, 3844, 58]-code), using
(199, 199+57, 1145305)-Net in Base 4 — Upper bound on s
There is no (199, 256, 1145306)-net in base 4, because
- 1 times m-reduction [i] would yield (199, 255, 1145306)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 3351 967064 702646 730695 586203 046127 715313 873630 168555 620409 864442 828859 955750 561454 509846 771957 446386 372632 355980 192802 806308 229033 982166 500654 507097 774640 > 4255 [i]