Best Known (189−28, 189, s)-Nets in Base 4
(189−28, 189, 18724)-Net over F4 — Constructive and digital
Digital (161, 189, 18724)-net over F4, using
- net defined by OOA [i] based on linear OOA(4189, 18724, F4, 28, 28) (dual of [(18724, 28), 524083, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(4189, 262136, F4, 28) (dual of [262136, 261947, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(4189, 262143, F4, 28) (dual of [262143, 261954, 29]-code), using
- 1 times truncation [i] based on linear OA(4190, 262144, F4, 29) (dual of [262144, 261954, 30]-code), using
- an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- 1 times truncation [i] based on linear OA(4190, 262144, F4, 29) (dual of [262144, 261954, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4189, 262143, F4, 28) (dual of [262143, 261954, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(4189, 262136, F4, 28) (dual of [262136, 261947, 29]-code), using
(189−28, 189, 108102)-Net over F4 — Digital
Digital (161, 189, 108102)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4189, 108102, F4, 2, 28) (dual of [(108102, 2), 216015, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4189, 131071, F4, 2, 28) (dual of [(131071, 2), 261953, 29]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4189, 262142, F4, 28) (dual of [262142, 261953, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(4189, 262143, F4, 28) (dual of [262143, 261954, 29]-code), using
- 1 times truncation [i] based on linear OA(4190, 262144, F4, 29) (dual of [262144, 261954, 30]-code), using
- an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- 1 times truncation [i] based on linear OA(4190, 262144, F4, 29) (dual of [262144, 261954, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4189, 262143, F4, 28) (dual of [262143, 261954, 29]-code), using
- OOA 2-folding [i] based on linear OA(4189, 262142, F4, 28) (dual of [262142, 261953, 29]-code), using
- discarding factors / shortening the dual code based on linear OOA(4189, 131071, F4, 2, 28) (dual of [(131071, 2), 261953, 29]-NRT-code), using
(189−28, 189, large)-Net in Base 4 — Upper bound on s
There is no (161, 189, large)-net in base 4, because
- 26 times m-reduction [i] would yield (161, 163, large)-net in base 4, but