Best Known (177−27, 177, s)-Nets in Base 4
(177−27, 177, 5055)-Net over F4 — Constructive and digital
Digital (150, 177, 5055)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 16, 14)-net over F4, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 3 and N(F) ≥ 14, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- digital (134, 161, 5041)-net over F4, using
- net defined by OOA [i] based on linear OOA(4161, 5041, F4, 27, 27) (dual of [(5041, 27), 135946, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4161, 65534, F4, 27) (dual of [65534, 65373, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4161, 65536, F4, 27) (dual of [65536, 65375, 28]-code), using
- an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- discarding factors / shortening the dual code based on linear OA(4161, 65536, F4, 27) (dual of [65536, 65375, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4161, 65534, F4, 27) (dual of [65534, 65373, 28]-code), using
- net defined by OOA [i] based on linear OOA(4161, 5041, F4, 27, 27) (dual of [(5041, 27), 135946, 28]-NRT-code), using
- digital (3, 16, 14)-net over F4, using
(177−27, 177, 58730)-Net over F4 — Digital
Digital (150, 177, 58730)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4177, 58730, F4, 27) (dual of [58730, 58553, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4177, 65556, F4, 27) (dual of [65556, 65379, 28]-code), using
- (u, u+v)-construction [i] based on
- linear OA(416, 19, F4, 13) (dual of [19, 3, 14]-code), using
- 2 times truncation [i] based on linear OA(418, 21, F4, 15) (dual of [21, 3, 16]-code), using
- Simplex code S(3,4) [i]
- 2 times truncation [i] based on linear OA(418, 21, F4, 15) (dual of [21, 3, 16]-code), using
- linear OA(4161, 65537, F4, 27) (dual of [65537, 65376, 28]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(416, 19, F4, 13) (dual of [19, 3, 14]-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(4177, 65556, F4, 27) (dual of [65556, 65379, 28]-code), using
(177−27, 177, large)-Net in Base 4 — Upper bound on s
There is no (150, 177, large)-net in base 4, because
- 25 times m-reduction [i] would yield (150, 152, large)-net in base 4, but