Best Known (189, 231, s)-Nets in Base 4
(189, 231, 1544)-Net over F4 — Constructive and digital
Digital (189, 231, 1544)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 21, 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 (168, 210, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 70, 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, 70, 513)-net over F64, using
- digital (0, 21, 5)-net over F4, using
(189, 231, 15191)-Net over F4 — Digital
Digital (189, 231, 15191)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4231, 15191, F4, 42) (dual of [15191, 14960, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(4231, 16406, F4, 42) (dual of [16406, 16175, 43]-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(46, 21, F4, 4) (dual of [21, 15, 5]-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(4231, 16406, F4, 42) (dual of [16406, 16175, 43]-code), using
(189, 231, large)-Net in Base 4 — Upper bound on s
There is no (189, 231, large)-net in base 4, because
- 40 times m-reduction [i] would yield (189, 191, large)-net in base 4, but