Best Known (242−18, 242, s)-Nets in Base 4
(242−18, 242, 3728782)-Net over F4 — Constructive and digital
Digital (224, 242, 3728782)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (25, 34, 514)-net over F4, using
- trace code for nets [i] based on digital (8, 17, 257)-net over F16, using
- base reduction for projective spaces (embedding PG(8,256) in PG(16,16)) for nets [i] based on digital (0, 9, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- base reduction for projective spaces (embedding PG(8,256) in PG(16,16)) for nets [i] based on digital (0, 9, 257)-net over F256, using
- trace code for nets [i] based on digital (8, 17, 257)-net over F16, using
- digital (190, 208, 3728268)-net over F4, using
- trace code for nets [i] based on digital (34, 52, 932067)-net over F256, using
- net defined by OOA [i] based on linear OOA(25652, 932067, F256, 18, 18) (dual of [(932067, 18), 16777154, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(25652, large, F256, 18) (dual of [large, large−52, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OA 9-folding and stacking [i] based on linear OA(25652, large, F256, 18) (dual of [large, large−52, 19]-code), using
- net defined by OOA [i] based on linear OOA(25652, 932067, F256, 18, 18) (dual of [(932067, 18), 16777154, 19]-NRT-code), using
- trace code for nets [i] based on digital (34, 52, 932067)-net over F256, using
- digital (25, 34, 514)-net over F4, using
(242−18, 242, large)-Net over F4 — Digital
Digital (224, 242, large)-net over F4, using
- t-expansion [i] based on digital (222, 242, large)-net over F4, using
- 5 times m-reduction [i] based on digital (222, 247, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4247, large, F4, 25) (dual of [large, large−247, 26]-code), using
- 30 times code embedding in larger space [i] based on linear OA(4217, large, F4, 25) (dual of [large, large−217, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 424−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 30 times code embedding in larger space [i] based on linear OA(4217, large, F4, 25) (dual of [large, large−217, 26]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4247, large, F4, 25) (dual of [large, large−247, 26]-code), using
- 5 times m-reduction [i] based on digital (222, 247, large)-net over F4, using
(242−18, 242, large)-Net in Base 4 — Upper bound on s
There is no (224, 242, large)-net in base 4, because
- 16 times m-reduction [i] would yield (224, 226, large)-net in base 4, but