Best Known (181, 213, s)-Nets in Base 4
(181, 213, 4113)-Net over F4 — Constructive and digital
Digital (181, 213, 4113)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (5, 21, 17)-net over F4, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 5 and N(F) ≥ 17, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- digital (160, 192, 4096)-net over F4, using
- net defined by OOA [i] based on linear OOA(4192, 4096, F4, 32, 32) (dual of [(4096, 32), 130880, 33]-NRT-code), using
- OA 16-folding and stacking [i] based on linear OA(4192, 65536, F4, 32) (dual of [65536, 65344, 33]-code), using
- 1 times truncation [i] based on linear OA(4193, 65537, F4, 33) (dual of [65537, 65344, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(4193, 65537, F4, 33) (dual of [65537, 65344, 34]-code), using
- OA 16-folding and stacking [i] based on linear OA(4192, 65536, F4, 32) (dual of [65536, 65344, 33]-code), using
- net defined by OOA [i] based on linear OOA(4192, 4096, F4, 32, 32) (dual of [(4096, 32), 130880, 33]-NRT-code), using
- digital (5, 21, 17)-net over F4, using
(181, 213, 65613)-Net over F4 — Digital
Digital (181, 213, 65613)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4213, 65613, F4, 32) (dual of [65613, 65400, 33]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4212, 65611, F4, 32) (dual of [65611, 65399, 33]-code), using
- construction X applied to Ce(32) ⊂ Ce(22) [i] based on
- linear OA(4193, 65536, F4, 33) (dual of [65536, 65343, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(4137, 65536, F4, 23) (dual of [65536, 65399, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(419, 75, F4, 8) (dual of [75, 56, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(419, 87, F4, 8) (dual of [87, 68, 9]-code), using
- construction X applied to Ce(32) ⊂ Ce(22) [i] based on
- linear OA(4212, 65612, F4, 31) (dual of [65612, 65400, 32]-code), using Gilbert–Varšamov bound and bm = 4212 > Vbs−1(k−1) = 2 491516 148176 583161 754533 376930 039343 273609 642617 689090 942386 863355 587601 404985 914567 139226 649891 556421 913112 673997 243819 444944 [i]
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(4212, 65611, F4, 32) (dual of [65611, 65399, 33]-code), using
- construction X with Varšamov bound [i] based on
(181, 213, large)-Net in Base 4 — Upper bound on s
There is no (181, 213, large)-net in base 4, because
- 30 times m-reduction [i] would yield (181, 183, large)-net in base 4, but