Best Known (204, 204+49, s)-Nets in Base 4
(204, 204+49, 1539)-Net over F4 — Constructive and digital
Digital (204, 253, 1539)-net over F4, using
- t-expansion [i] based on digital (200, 253, 1539)-net over F4, using
- 5 times m-reduction [i] based on digital (200, 258, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 86, 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, 86, 513)-net over F64, using
- 5 times m-reduction [i] based on digital (200, 258, 1539)-net over F4, using
(204, 204+49, 10315)-Net over F4 — Digital
Digital (204, 253, 10315)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4253, 10315, F4, 49) (dual of [10315, 10062, 50]-code), using
- discarding factors / shortening the dual code based on linear OA(4253, 16384, F4, 49) (dual of [16384, 16131, 50]-code), using
- an extension Ce(48) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,48], and designed minimum distance d ≥ |I|+1 = 49 [i]
- discarding factors / shortening the dual code based on linear OA(4253, 16384, F4, 49) (dual of [16384, 16131, 50]-code), using
(204, 204+49, 6852833)-Net in Base 4 — Upper bound on s
There is no (204, 253, 6852834)-net in base 4, because
- 1 times m-reduction [i] would yield (204, 252, 6852834)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 52 374415 009686 871431 298603 284367 213741 532882 145201 701308 826787 139305 596002 601966 517509 788665 368407 946183 052277 369383 344254 479142 063547 122640 550868 326196 > 4252 [i]