Best Known (174−27, 174, s)-Nets in Base 4
(174−27, 174, 5046)-Net over F4 — Constructive and digital
Digital (147, 174, 5046)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 13, 5)-net over F4, using
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 0 and N(F) ≥ 5, using
- the rational function field F4(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- digital (134, 161, 5041)-net over F4, using
- net defined by OOA [i] based on linear OOA(4161, 5041, F4, 27, 27) (dual of [(5041, 27), 135946, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4161, 65534, F4, 27) (dual of [65534, 65373, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4161, 65536, F4, 27) (dual of [65536, 65375, 28]-code), using
- an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- discarding factors / shortening the dual code based on linear OA(4161, 65536, F4, 27) (dual of [65536, 65375, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4161, 65534, F4, 27) (dual of [65534, 65373, 28]-code), using
- net defined by OOA [i] based on linear OOA(4161, 5041, F4, 27, 27) (dual of [(5041, 27), 135946, 28]-NRT-code), using
- digital (0, 13, 5)-net over F4, using
(174−27, 174, 49726)-Net over F4 — Digital
Digital (147, 174, 49726)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4174, 49726, F4, 27) (dual of [49726, 49552, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4174, 65589, F4, 27) (dual of [65589, 65415, 28]-code), using
- 3 times code embedding in larger space [i] based on linear OA(4171, 65586, F4, 27) (dual of [65586, 65415, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(20) [i] based on
- linear OA(4161, 65536, F4, 27) (dual of [65536, 65375, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(4121, 65536, F4, 21) (dual of [65536, 65415, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(410, 50, F4, 5) (dual of [50, 40, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to Ce(26) ⊂ Ce(20) [i] based on
- 3 times code embedding in larger space [i] based on linear OA(4171, 65586, F4, 27) (dual of [65586, 65415, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4174, 65589, F4, 27) (dual of [65589, 65415, 28]-code), using
(174−27, 174, large)-Net in Base 4 — Upper bound on s
There is no (147, 174, large)-net in base 4, because
- 25 times m-reduction [i] would yield (147, 149, large)-net in base 4, but