Best Known (159−25, 159, s)-Nets in Base 4
(159−25, 159, 5471)-Net over F4 — Constructive and digital
Digital (134, 159, 5471)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (2, 14, 10)-net over F4, using
- net from sequence [i] based on digital (2, 9)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 2 and N(F) ≥ 10, using
- net from sequence [i] based on digital (2, 9)-sequence over F4, using
- digital (120, 145, 5461)-net over F4, using
- net defined by OOA [i] based on linear OOA(4145, 5461, F4, 25, 25) (dual of [(5461, 25), 136380, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4145, 65533, F4, 25) (dual of [65533, 65388, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4145, 65536, F4, 25) (dual of [65536, 65391, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(4145, 65536, F4, 25) (dual of [65536, 65391, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4145, 65533, F4, 25) (dual of [65533, 65388, 26]-code), using
- net defined by OOA [i] based on linear OOA(4145, 5461, F4, 25, 25) (dual of [(5461, 25), 136380, 26]-NRT-code), using
- digital (2, 14, 10)-net over F4, using
(159−25, 159, 42958)-Net over F4 — Digital
Digital (134, 159, 42958)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4159, 42958, F4, 25) (dual of [42958, 42799, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4159, 65590, F4, 25) (dual of [65590, 65431, 26]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4158, 65589, F4, 25) (dual of [65589, 65431, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(17) [i] based on
- linear OA(4145, 65536, F4, 25) (dual of [65536, 65391, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(4105, 65536, F4, 18) (dual of [65536, 65431, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(413, 53, F4, 6) (dual of [53, 40, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- construction X applied to Ce(24) ⊂ Ce(17) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4158, 65589, F4, 25) (dual of [65589, 65431, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4159, 65590, F4, 25) (dual of [65590, 65431, 26]-code), using
(159−25, 159, large)-Net in Base 4 — Upper bound on s
There is no (134, 159, large)-net in base 4, because
- 23 times m-reduction [i] would yield (134, 136, large)-net in base 4, but