Best Known (144, 144+41, s)-Nets in Base 4
(144, 144+41, 1048)-Net over F4 — Constructive and digital
Digital (144, 185, 1048)-net over F4, using
- 41 times duplication [i] based on digital (143, 184, 1048)-net over F4, using
- trace code for nets [i] based on digital (5, 46, 262)-net over F256, using
- net from sequence [i] based on digital (5, 261)-sequence over F256, using
- trace code for nets [i] based on digital (5, 46, 262)-net over F256, using
(144, 144+41, 3524)-Net over F4 — Digital
Digital (144, 185, 3524)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4185, 3524, F4, 41) (dual of [3524, 3339, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(4185, 4113, F4, 41) (dual of [4113, 3928, 42]-code), using
- construction X applied to C([0,20]) ⊂ C([0,18]) [i] based on
- linear OA(4181, 4097, F4, 41) (dual of [4097, 3916, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,20], and minimum distance d ≥ |{−20,−19,…,20}|+1 = 42 (BCH-bound) [i]
- linear OA(4169, 4097, F4, 37) (dual of [4097, 3928, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(44, 16, F4, 3) (dual of [16, 12, 4]-code or 16-cap in PG(3,4)), using
- construction X applied to C([0,20]) ⊂ C([0,18]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4185, 4113, F4, 41) (dual of [4113, 3928, 42]-code), using
(144, 144+41, 957479)-Net in Base 4 — Upper bound on s
There is no (144, 185, 957480)-net in base 4, because
- 1 times m-reduction [i] would yield (144, 184, 957480)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 601 230986 719185 519895 239923 114948 849649 754752 406695 791941 264230 006456 218546 461521 488145 168889 486305 929241 165466 > 4184 [i]