Best Known (14, 19, s)-Nets in Base 4
(14, 19, 2047)-Net over F4 — Constructive and digital
Digital (14, 19, 2047)-net over F4, using
- net defined by OOA [i] based on linear OOA(419, 2047, F4, 5, 5) (dual of [(2047, 5), 10216, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(419, 4095, F4, 5) (dual of [4095, 4076, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(419, 4096, F4, 5) (dual of [4096, 4077, 6]-code), using
- an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(419, 4096, F4, 5) (dual of [4096, 4077, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(419, 4095, F4, 5) (dual of [4095, 4076, 6]-code), using
(14, 19, 2479)-Net over F4 — Digital
Digital (14, 19, 2479)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(419, 2479, F4, 5) (dual of [2479, 2460, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(419, 4096, F4, 5) (dual of [4096, 4077, 6]-code), using
- an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(419, 4096, F4, 5) (dual of [4096, 4077, 6]-code), using
(14, 19, 123574)-Net in Base 4 — Upper bound on s
There is no (14, 19, 123575)-net in base 4, because
- 1 times m-reduction [i] would yield (14, 18, 123575)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 68719 810351 > 418 [i]