Best Known (181−28, 181, s)-Nets in Base 4
(181−28, 181, 4684)-Net over F4 — Constructive and digital
Digital (153, 181, 4684)-net over F4, using
- 42 times duplication [i] based on digital (151, 179, 4684)-net over F4, using
- t-expansion [i] based on digital (150, 179, 4684)-net over F4, using
- net defined by OOA [i] based on linear OOA(4179, 4684, F4, 29, 29) (dual of [(4684, 29), 135657, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4179, 65577, F4, 29) (dual of [65577, 65398, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4179, 65578, F4, 29) (dual of [65578, 65399, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- linear OA(4169, 65536, F4, 29) (dual of [65536, 65367, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- 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(410, 42, F4, 5) (dual of [42, 32, 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(28) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(4179, 65578, F4, 29) (dual of [65578, 65399, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4179, 65577, F4, 29) (dual of [65577, 65398, 30]-code), using
- net defined by OOA [i] based on linear OOA(4179, 4684, F4, 29, 29) (dual of [(4684, 29), 135657, 30]-NRT-code), using
- t-expansion [i] based on digital (150, 179, 4684)-net over F4, using
(181−28, 181, 51773)-Net over F4 — Digital
Digital (153, 181, 51773)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4181, 51773, F4, 28) (dual of [51773, 51592, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(4181, 65588, F4, 28) (dual of [65588, 65407, 29]-code), using
- 1 times truncation [i] based on linear OA(4182, 65589, F4, 29) (dual of [65589, 65407, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(21) [i] based on
- linear OA(4169, 65536, F4, 29) (dual of [65536, 65367, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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(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(28) ⊂ Ce(21) [i] based on
- 1 times truncation [i] based on linear OA(4182, 65589, F4, 29) (dual of [65589, 65407, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4181, 65588, F4, 28) (dual of [65588, 65407, 29]-code), using
(181−28, 181, large)-Net in Base 4 — Upper bound on s
There is no (153, 181, large)-net in base 4, because
- 26 times m-reduction [i] would yield (153, 155, large)-net in base 4, but