Best Known (163−27, 163, s)-Nets in Base 4
(163−27, 163, 5042)-Net over F4 — Constructive and digital
Digital (136, 163, 5042)-net over F4, using
- 41 times duplication [i] based on digital (135, 162, 5042)-net over F4, using
- net defined by OOA [i] based on linear OOA(4162, 5042, F4, 27, 27) (dual of [(5042, 27), 135972, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4162, 65547, F4, 27) (dual of [65547, 65385, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4162, 65554, F4, 27) (dual of [65554, 65392, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(4161, 65537, F4, 27) (dual of [65537, 65376, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (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(41, 17, F4, 1) (dual of [17, 16, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4162, 65554, F4, 27) (dual of [65554, 65392, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4162, 65547, F4, 27) (dual of [65547, 65385, 28]-code), using
- net defined by OOA [i] based on linear OOA(4162, 5042, F4, 27, 27) (dual of [(5042, 27), 135972, 28]-NRT-code), using
(163−27, 163, 32777)-Net over F4 — Digital
Digital (136, 163, 32777)-net over F4, using
- 41 times duplication [i] based on digital (135, 162, 32777)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4162, 32777, F4, 2, 27) (dual of [(32777, 2), 65392, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4162, 65554, F4, 27) (dual of [65554, 65392, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(4161, 65537, F4, 27) (dual of [65537, 65376, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (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(41, 17, F4, 1) (dual of [17, 16, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- OOA 2-folding [i] based on linear OA(4162, 65554, F4, 27) (dual of [65554, 65392, 28]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4162, 32777, F4, 2, 27) (dual of [(32777, 2), 65392, 28]-NRT-code), using
(163−27, 163, large)-Net in Base 4 — Upper bound on s
There is no (136, 163, large)-net in base 4, because
- 25 times m-reduction [i] would yield (136, 138, large)-net in base 4, but