Best Known (227−41, 227, s)-Nets in Base 4
(227−41, 227, 1544)-Net over F4 — Constructive and digital
Digital (186, 227, 1544)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 20, 5)-net over F4, using
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 0 and N(F) ≥ 5, using
- the rational function field F4(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- digital (166, 207, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 69, 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, 69, 513)-net over F64, using
- digital (0, 20, 5)-net over F4, using
(227−41, 227, 15787)-Net over F4 — Digital
Digital (186, 227, 15787)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4227, 15787, F4, 41) (dual of [15787, 15560, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(4227, 16443, F4, 41) (dual of [16443, 16216, 42]-code), using
- construction X applied to C([0,20]) ⊂ C([0,16]) [i] based on
- linear OA(4211, 16385, F4, 41) (dual of [16385, 16174, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,20], and minimum distance d ≥ |{−20,−19,…,20}|+1 = 42 (BCH-bound) [i]
- linear OA(4169, 16385, F4, 33) (dual of [16385, 16216, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(416, 58, F4, 7) (dual of [58, 42, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(416, 63, F4, 7) (dual of [63, 47, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(416, 63, F4, 7) (dual of [63, 47, 8]-code), using
- construction X applied to C([0,20]) ⊂ C([0,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4227, 16443, F4, 41) (dual of [16443, 16216, 42]-code), using
(227−41, 227, large)-Net in Base 4 — Upper bound on s
There is no (186, 227, large)-net in base 4, because
- 39 times m-reduction [i] would yield (186, 188, large)-net in base 4, but