Best Known (23, 23+5, s)-Nets in Base 4
(23, 23+5, 131071)-Net over F4 — Constructive and digital
Digital (23, 28, 131071)-net over F4, using
- net defined by OOA [i] based on linear OOA(428, 131071, F4, 5, 5) (dual of [(131071, 5), 655327, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(428, 262143, F4, 5) (dual of [262143, 262115, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(428, 262144, F4, 5) (dual of [262144, 262116, 6]-code), using
- an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(428, 262144, F4, 5) (dual of [262144, 262116, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(428, 262143, F4, 5) (dual of [262143, 262115, 6]-code), using
(23, 23+5, 158781)-Net over F4 — Digital
Digital (23, 28, 158781)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(428, 158781, F4, 5) (dual of [158781, 158753, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(428, 262144, F4, 5) (dual of [262144, 262116, 6]-code), using
- an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(428, 262144, F4, 5) (dual of [262144, 262116, 6]-code), using
(23, 23+5, large)-Net in Base 4 — Upper bound on s
There is no (23, 28, large)-net in base 4, because
- 3 times m-reduction [i] would yield (23, 25, large)-net in base 4, but