Best Known (176, 176+42, s)-Nets in Base 4
(176, 176+42, 1539)-Net over F4 — Constructive and digital
Digital (176, 218, 1539)-net over F4, using
- 4 times m-reduction [i] based on digital (176, 222, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 74, 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, 74, 513)-net over F64, using
(176, 176+42, 9670)-Net over F4 — Digital
Digital (176, 218, 9670)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4218, 9670, F4, 42) (dual of [9670, 9452, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(4218, 16384, F4, 42) (dual of [16384, 16166, 43]-code), using
- an extension Ce(41) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,41], and designed minimum distance d ≥ |I|+1 = 42 [i]
- discarding factors / shortening the dual code based on linear OA(4218, 16384, F4, 42) (dual of [16384, 16166, 43]-code), using
(176, 176+42, 5144296)-Net in Base 4 — Upper bound on s
There is no (176, 218, 5144297)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 177451 343960 488318 416981 137115 578630 176224 126373 707252 072697 290473 059927 249329 379841 212273 718588 732868 829748 660874 736951 930440 372352 > 4218 [i]