Best Known (231−44, 231, s)-Nets in Base 4
(231−44, 231, 1539)-Net over F4 — Constructive and digital
Digital (187, 231, 1539)-net over F4, using
- t-expansion [i] based on digital (186, 231, 1539)-net over F4, using
- 6 times m-reduction [i] based on digital (186, 237, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 79, 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, 79, 513)-net over F64, using
- 6 times m-reduction [i] based on digital (186, 237, 1539)-net over F4, using
(231−44, 231, 10873)-Net over F4 — Digital
Digital (187, 231, 10873)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4231, 10873, F4, 44) (dual of [10873, 10642, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(4231, 16383, F4, 44) (dual of [16383, 16152, 45]-code), using
- 1 times truncation [i] based on linear OA(4232, 16384, F4, 45) (dual of [16384, 16152, 46]-code), using
- an extension Ce(44) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,44], and designed minimum distance d ≥ |I|+1 = 45 [i]
- 1 times truncation [i] based on linear OA(4232, 16384, F4, 45) (dual of [16384, 16152, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(4231, 16383, F4, 44) (dual of [16383, 16152, 45]-code), using
(231−44, 231, 6329369)-Net in Base 4 — Upper bound on s
There is no (187, 231, 6329370)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 11 908541 787700 371599 671576 305110 295733 106536 611078 033445 705684 300066 340155 592670 304799 472664 491149 466903 623124 763723 887208 589092 152401 845324 > 4231 [i]