Best Known (209−32, 209, s)-Nets in Base 4
(209−32, 209, 4105)-Net over F4 — Constructive and digital
Digital (177, 209, 4105)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 17, 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 (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 (1, 17, 9)-net over F4, using
(209−32, 209, 59948)-Net over F4 — Digital
Digital (177, 209, 59948)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4209, 59948, F4, 32) (dual of [59948, 59739, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(4209, 65601, F4, 32) (dual of [65601, 65392, 33]-code), using
- strength reduction [i] based on linear OA(4209, 65601, F4, 33) (dual of [65601, 65392, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,12]) [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]
- linear OA(4145, 65537, F4, 25) (dual of [65537, 65392, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (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([0,16]) ⊂ C([0,12]) [i] based on
- strength reduction [i] based on linear OA(4209, 65601, F4, 33) (dual of [65601, 65392, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(4209, 65601, F4, 32) (dual of [65601, 65392, 33]-code), using
(209−32, 209, large)-Net in Base 4 — Upper bound on s
There is no (177, 209, large)-net in base 4, because
- 30 times m-reduction [i] would yield (177, 179, large)-net in base 4, but