Best Known (72−16, 72, s)-Nets in Base 4
(72−16, 72, 1036)-Net over F4 — Constructive and digital
Digital (56, 72, 1036)-net over F4, using
- trace code for nets [i] based on digital (2, 18, 259)-net over F256, using
- net from sequence [i] based on digital (2, 258)-sequence over F256, using
(72−16, 72, 2268)-Net over F4 — Digital
Digital (56, 72, 2268)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(472, 2268, F4, 16) (dual of [2268, 2196, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(472, 4096, F4, 16) (dual of [4096, 4024, 17]-code), using
- 1 times truncation [i] based on linear OA(473, 4097, F4, 17) (dual of [4097, 4024, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(473, 4097, F4, 17) (dual of [4097, 4024, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(472, 4096, F4, 16) (dual of [4096, 4024, 17]-code), using
(72−16, 72, 328927)-Net in Base 4 — Upper bound on s
There is no (56, 72, 328928)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 22 300849 390372 668778 816063 823679 858277 107085 > 472 [i]