Best Known (109−17, 109, s)-Nets in Base 4
(109−17, 109, 32768)-Net over F4 — Constructive and digital
Digital (92, 109, 32768)-net over F4, using
- net defined by OOA [i] based on linear OOA(4109, 32768, F4, 17, 17) (dual of [(32768, 17), 556947, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(4109, 262145, F4, 17) (dual of [262145, 262036, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(4109, 262145, F4, 17) (dual of [262145, 262036, 18]-code), using
(109−17, 109, 87381)-Net over F4 — Digital
Digital (92, 109, 87381)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4109, 87381, F4, 3, 17) (dual of [(87381, 3), 262034, 18]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4109, 262143, F4, 17) (dual of [262143, 262034, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(4109, 262144, F4, 17) (dual of [262144, 262035, 18]-code), using
- an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(4109, 262144, F4, 17) (dual of [262144, 262035, 18]-code), using
- OOA 3-folding [i] based on linear OA(4109, 262143, F4, 17) (dual of [262143, 262034, 18]-code), using
(109−17, 109, large)-Net in Base 4 — Upper bound on s
There is no (92, 109, large)-net in base 4, because
- 15 times m-reduction [i] would yield (92, 94, large)-net in base 4, but