Best Known (182−29, 182, s)-Nets in Base 4
(182−29, 182, 4684)-Net over F4 — Constructive and digital
Digital (153, 182, 4684)-net over F4, using
- 43 times duplication [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
(182−29, 182, 39544)-Net over F4 — Digital
Digital (153, 182, 39544)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4182, 39544, F4, 29) (dual of [39544, 39362, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4182, 65574, F4, 29) (dual of [65574, 65392, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- linear OA(4177, 65537, F4, 29) (dual of [65537, 65360, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(4145, 65537, F4, 25) (dual of [65537, 65392, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(45, 37, F4, 3) (dual of [37, 32, 4]-code or 37-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4182, 65574, F4, 29) (dual of [65574, 65392, 30]-code), using
(182−29, 182, large)-Net in Base 4 — Upper bound on s
There is no (153, 182, large)-net in base 4, because
- 27 times m-reduction [i] would yield (153, 155, large)-net in base 4, but