Best Known (227−35, 227, s)-Nets in Base 4
(227−35, 227, 3864)-Net over F4 — Constructive and digital
Digital (192, 227, 3864)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 18, 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 (174, 209, 3855)-net over F4, using
- net defined by OOA [i] based on linear OOA(4209, 3855, F4, 35, 35) (dual of [(3855, 35), 134716, 36]-NRT-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(4209, 65536, F4, 35) (dual of [65536, 65327, 36]-code), using
- an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- OOA 17-folding and stacking with additional row [i] based on linear OA(4209, 65536, F4, 35) (dual of [65536, 65327, 36]-code), using
- net defined by OOA [i] based on linear OOA(4209, 3855, F4, 35, 35) (dual of [(3855, 35), 134716, 36]-NRT-code), using
- digital (1, 18, 9)-net over F4, using
(227−35, 227, 58242)-Net over F4 — Digital
Digital (192, 227, 58242)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4227, 58242, F4, 35) (dual of [58242, 58015, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(4227, 65603, F4, 35) (dual of [65603, 65376, 36]-code), using
- 2 times code embedding in larger space [i] based on linear OA(4225, 65601, F4, 35) (dual of [65601, 65376, 36]-code), using
- construction X applied to C([0,17]) ⊂ C([0,13]) [i] based on
- linear OA(4209, 65537, F4, 35) (dual of [65537, 65328, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (BCH-bound) [i]
- linear OA(4161, 65537, F4, 27) (dual of [65537, 65376, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(416, 64, F4, 7) (dual of [64, 48, 8]-code), using
- an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- construction X applied to C([0,17]) ⊂ C([0,13]) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(4225, 65601, F4, 35) (dual of [65601, 65376, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(4227, 65603, F4, 35) (dual of [65603, 65376, 36]-code), using
(227−35, 227, large)-Net in Base 4 — Upper bound on s
There is no (192, 227, large)-net in base 4, because
- 33 times m-reduction [i] would yield (192, 194, large)-net in base 4, but