Best Known (166, 166+24, s)-Nets in Base 4
(166, 166+24, 87385)-Net over F4 — Constructive and digital
Digital (166, 190, 87385)-net over F4, using
- 1 times m-reduction [i] based on digital (166, 191, 87385)-net over F4, using
- net defined by OOA [i] based on linear OOA(4191, 87385, F4, 25, 25) (dual of [(87385, 25), 2184434, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4191, 1048621, F4, 25) (dual of [1048621, 1048430, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4191, 1048627, F4, 25) (dual of [1048627, 1048436, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(4181, 1048577, F4, 25) (dual of [1048577, 1048396, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(4141, 1048577, F4, 19) (dual of [1048577, 1048436, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [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 C([0,12]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4191, 1048627, F4, 25) (dual of [1048627, 1048436, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4191, 1048621, F4, 25) (dual of [1048621, 1048430, 26]-code), using
- net defined by OOA [i] based on linear OOA(4191, 87385, F4, 25, 25) (dual of [(87385, 25), 2184434, 26]-NRT-code), using
(166, 166+24, 524313)-Net over F4 — Digital
Digital (166, 190, 524313)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4190, 524313, F4, 2, 24) (dual of [(524313, 2), 1048436, 25]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4190, 1048626, F4, 24) (dual of [1048626, 1048436, 25]-code), using
- 1 times truncation [i] based on linear OA(4191, 1048627, F4, 25) (dual of [1048627, 1048436, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(4181, 1048577, F4, 25) (dual of [1048577, 1048396, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(4141, 1048577, F4, 19) (dual of [1048577, 1048436, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [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 C([0,12]) ⊂ C([0,9]) [i] based on
- 1 times truncation [i] based on linear OA(4191, 1048627, F4, 25) (dual of [1048627, 1048436, 26]-code), using
- OOA 2-folding [i] based on linear OA(4190, 1048626, F4, 24) (dual of [1048626, 1048436, 25]-code), using
(166, 166+24, large)-Net in Base 4 — Upper bound on s
There is no (166, 190, large)-net in base 4, because
- 22 times m-reduction [i] would yield (166, 168, large)-net in base 4, but