Best Known (192, 192+43, s)-Nets in Base 4
(192, 192+43, 1548)-Net over F4 — Constructive and digital
Digital (192, 235, 1548)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 22, 9)-net over F4, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- the Hermitian function field over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- 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
- digital (1, 22, 9)-net over F4, using
(192, 192+43, 14654)-Net over F4 — Digital
Digital (192, 235, 14654)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4235, 14654, F4, 43) (dual of [14654, 14419, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(4235, 16423, F4, 43) (dual of [16423, 16188, 44]-code), using
- construction X applied to C([0,21]) ⊂ C([0,18]) [i] based on
- linear OA(4225, 16385, F4, 43) (dual of [16385, 16160, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,21], and minimum distance d ≥ |{−21,−20,…,21}|+1 = 44 (BCH-bound) [i]
- linear OA(4197, 16385, F4, 37) (dual of [16385, 16188, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(410, 38, F4, 5) (dual of [38, 28, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to C([0,21]) ⊂ C([0,18]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4235, 16423, F4, 43) (dual of [16423, 16188, 44]-code), using
(192, 192+43, large)-Net in Base 4 — Upper bound on s
There is no (192, 235, large)-net in base 4, because
- 41 times m-reduction [i] would yield (192, 194, large)-net in base 4, but