Best Known (128, 128+23, s)-Nets in Base 4
(128, 128+23, 5972)-Net over F4 — Constructive and digital
Digital (128, 151, 5972)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 14, 14)-net over F4, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 3 and N(F) ≥ 14, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- digital (114, 137, 5958)-net over F4, using
- net defined by OOA [i] based on linear OOA(4137, 5958, F4, 23, 23) (dual of [(5958, 23), 136897, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4137, 65539, F4, 23) (dual of [65539, 65402, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4137, 65544, F4, 23) (dual of [65544, 65407, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- 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(4129, 65536, F4, 22) (dual of [65536, 65407, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(40, 8, F4, 0) (dual of [8, 8, 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
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(4137, 65544, F4, 23) (dual of [65544, 65407, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4137, 65539, F4, 23) (dual of [65539, 65402, 24]-code), using
- net defined by OOA [i] based on linear OOA(4137, 5958, F4, 23, 23) (dual of [(5958, 23), 136897, 24]-NRT-code), using
- digital (3, 14, 14)-net over F4, using
(128, 128+23, 57766)-Net over F4 — Digital
Digital (128, 151, 57766)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4151, 57766, F4, 23) (dual of [57766, 57615, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4151, 65590, F4, 23) (dual of [65590, 65439, 24]-code), using
- 4 times code embedding in larger space [i] based on linear OA(4147, 65586, F4, 23) (dual of [65586, 65439, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- 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(497, 65536, F4, 17) (dual of [65536, 65439, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(22) ⊂ Ce(16) [i] based on
- 4 times code embedding in larger space [i] based on linear OA(4147, 65586, F4, 23) (dual of [65586, 65439, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4151, 65590, F4, 23) (dual of [65590, 65439, 24]-code), using
(128, 128+23, large)-Net in Base 4 — Upper bound on s
There is no (128, 151, large)-net in base 4, because
- 21 times m-reduction [i] would yield (128, 130, large)-net in base 4, but