Best Known (209−22, 209, s)-Nets in Base 4
(209−22, 209, 762617)-Net over F4 — Constructive and digital
Digital (187, 209, 762617)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (5, 16, 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 (171, 193, 762600)-net over F4, using
- net defined by OOA [i] based on linear OOA(4193, 762600, F4, 22, 22) (dual of [(762600, 22), 16777007, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(4193, 8388600, F4, 22) (dual of [8388600, 8388407, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4193, large, F4, 22) (dual of [large, large−193, 23]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(4193, large, F4, 22) (dual of [large, large−193, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(4193, 8388600, F4, 22) (dual of [8388600, 8388407, 23]-code), using
- net defined by OOA [i] based on linear OOA(4193, 762600, F4, 22, 22) (dual of [(762600, 22), 16777007, 23]-NRT-code), using
- digital (5, 16, 17)-net over F4, using
(209−22, 209, 5053678)-Net over F4 — Digital
Digital (187, 209, 5053678)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4209, 5053678, F4, 22) (dual of [5053678, 5053469, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4209, large, F4, 22) (dual of [large, large−209, 23]-code), using
- 16 times code embedding in larger space [i] based on linear OA(4193, large, F4, 22) (dual of [large, large−193, 23]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- 16 times code embedding in larger space [i] based on linear OA(4193, large, F4, 22) (dual of [large, large−193, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4209, large, F4, 22) (dual of [large, large−209, 23]-code), using
(209−22, 209, large)-Net in Base 4 — Upper bound on s
There is no (187, 209, large)-net in base 4, because
- 20 times m-reduction [i] would yield (187, 189, large)-net in base 4, but