Best Known (219, 219+33, s)-Nets in Base 4
(219, 219+33, 65539)-Net over F4 — Constructive and digital
Digital (219, 252, 65539)-net over F4, using
- 41 times duplication [i] based on digital (218, 251, 65539)-net over F4, using
- net defined by OOA [i] based on linear OOA(4251, 65539, F4, 33, 33) (dual of [(65539, 33), 2162536, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(4251, 1048625, F4, 33) (dual of [1048625, 1048374, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(4251, 1048627, F4, 33) (dual of [1048627, 1048376, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,13]) [i] based on
- linear OA(4241, 1048577, F4, 33) (dual of [1048577, 1048336, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(4201, 1048577, F4, 27) (dual of [1048577, 1048376, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (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([0,16]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4251, 1048627, F4, 33) (dual of [1048627, 1048376, 34]-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(4251, 1048625, F4, 33) (dual of [1048625, 1048374, 34]-code), using
- net defined by OOA [i] based on linear OOA(4251, 65539, F4, 33, 33) (dual of [(65539, 33), 2162536, 34]-NRT-code), using
(219, 219+33, 417647)-Net over F4 — Digital
Digital (219, 252, 417647)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4252, 417647, F4, 2, 33) (dual of [(417647, 2), 835042, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4252, 524314, F4, 2, 33) (dual of [(524314, 2), 1048376, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4252, 1048628, F4, 33) (dual of [1048628, 1048376, 34]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4251, 1048627, F4, 33) (dual of [1048627, 1048376, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,13]) [i] based on
- linear OA(4241, 1048577, F4, 33) (dual of [1048577, 1048336, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(4201, 1048577, F4, 27) (dual of [1048577, 1048376, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (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([0,16]) ⊂ C([0,13]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4251, 1048627, F4, 33) (dual of [1048627, 1048376, 34]-code), using
- OOA 2-folding [i] based on linear OA(4252, 1048628, F4, 33) (dual of [1048628, 1048376, 34]-code), using
- discarding factors / shortening the dual code based on linear OOA(4252, 524314, F4, 2, 33) (dual of [(524314, 2), 1048376, 34]-NRT-code), using
(219, 219+33, large)-Net in Base 4 — Upper bound on s
There is no (219, 252, large)-net in base 4, because
- 31 times m-reduction [i] would yield (219, 221, large)-net in base 4, but