Best Known (193−33, 193, s)-Nets in Base 4
(193−33, 193, 4096)-Net over F4 — Constructive and digital
Digital (160, 193, 4096)-net over F4, using
- net defined by OOA [i] based on linear OOA(4193, 4096, F4, 33, 33) (dual of [(4096, 33), 134975, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(4193, 65537, F4, 33) (dual of [65537, 65344, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- OOA 16-folding and stacking with additional row [i] based on linear OA(4193, 65537, F4, 33) (dual of [65537, 65344, 34]-code), using
(193−33, 193, 27315)-Net over F4 — Digital
Digital (160, 193, 27315)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4193, 27315, F4, 2, 33) (dual of [(27315, 2), 54437, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4193, 32768, F4, 2, 33) (dual of [(32768, 2), 65343, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4193, 65536, F4, 33) (dual of [65536, 65343, 34]-code), using
- an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- OOA 2-folding [i] based on linear OA(4193, 65536, F4, 33) (dual of [65536, 65343, 34]-code), using
- discarding factors / shortening the dual code based on linear OOA(4193, 32768, F4, 2, 33) (dual of [(32768, 2), 65343, 34]-NRT-code), using
(193−33, 193, large)-Net in Base 4 — Upper bound on s
There is no (160, 193, large)-net in base 4, because
- 31 times m-reduction [i] would yield (160, 162, large)-net in base 4, but