Best Known (40, 49, s)-Nets in Base 4
(40, 49, 16384)-Net over F4 — Constructive and digital
Digital (40, 49, 16384)-net over F4, using
- net defined by OOA [i] based on linear OOA(449, 16384, F4, 9, 9) (dual of [(16384, 9), 147407, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(449, 65537, F4, 9) (dual of [65537, 65488, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(449, 65537, F4, 9) (dual of [65537, 65488, 10]-code), using
(40, 49, 32768)-Net over F4 — Digital
Digital (40, 49, 32768)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(449, 32768, F4, 2, 9) (dual of [(32768, 2), 65487, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(449, 65536, F4, 9) (dual of [65536, 65487, 10]-code), using
- an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- OOA 2-folding [i] based on linear OA(449, 65536, F4, 9) (dual of [65536, 65487, 10]-code), using
(40, 49, large)-Net in Base 4 — Upper bound on s
There is no (40, 49, large)-net in base 4, because
- 7 times m-reduction [i] would yield (40, 42, large)-net in base 4, but