Best Known (211−40, 211, s)-Nets in Base 4
(211−40, 211, 1539)-Net over F4 — Constructive and digital
Digital (171, 211, 1539)-net over F4, using
- t-expansion [i] based on digital (170, 211, 1539)-net over F4, using
- 2 times m-reduction [i] based on digital (170, 213, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 71, 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, 71, 513)-net over F64, using
- 2 times m-reduction [i] based on digital (170, 213, 1539)-net over F4, using
(211−40, 211, 10607)-Net over F4 — Digital
Digital (171, 211, 10607)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4211, 10607, F4, 40) (dual of [10607, 10396, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(4211, 16391, F4, 40) (dual of [16391, 16180, 41]-code), using
- 1 times truncation [i] based on linear OA(4212, 16392, F4, 41) (dual of [16392, 16180, 42]-code), using
- construction X applied to Ce(40) ⊂ Ce(38) [i] based on
- linear OA(4211, 16384, F4, 41) (dual of [16384, 16173, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(4204, 16384, F4, 39) (dual of [16384, 16180, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(41, 8, F4, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(40) ⊂ Ce(38) [i] based on
- 1 times truncation [i] based on linear OA(4212, 16392, F4, 41) (dual of [16392, 16180, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(4211, 16391, F4, 40) (dual of [16391, 16180, 41]-code), using
(211−40, 211, 6221810)-Net in Base 4 — Upper bound on s
There is no (171, 211, 6221811)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 10 830770 038386 904639 827504 214308 711923 578502 152871 200363 580314 621306 837564 264783 124690 063154 038773 304041 139942 856972 488727 171976 > 4211 [i]