Best Known (221−49, 221, s)-Nets in Base 4
(221−49, 221, 1052)-Net over F4 — Constructive and digital
Digital (172, 221, 1052)-net over F4, using
- 41 times duplication [i] based on digital (171, 220, 1052)-net over F4, using
- trace code for nets [i] based on digital (6, 55, 263)-net over F256, using
- net from sequence [i] based on digital (6, 262)-sequence over F256, using
- trace code for nets [i] based on digital (6, 55, 263)-net over F256, using
(221−49, 221, 3991)-Net over F4 — Digital
Digital (172, 221, 3991)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4221, 3991, F4, 49) (dual of [3991, 3770, 50]-code), using
- discarding factors / shortening the dual code based on linear OA(4221, 4113, F4, 49) (dual of [4113, 3892, 50]-code), using
- construction X applied to C([0,24]) ⊂ C([0,22]) [i] based on
- linear OA(4217, 4097, F4, 49) (dual of [4097, 3880, 50]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,24], and minimum distance d ≥ |{−24,−23,…,24}|+1 = 50 (BCH-bound) [i]
- linear OA(4205, 4097, F4, 45) (dual of [4097, 3892, 46]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,22], and minimum distance d ≥ |{−22,−21,…,22}|+1 = 46 (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([0,24]) ⊂ C([0,22]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4221, 4113, F4, 49) (dual of [4113, 3892, 50]-code), using
(221−49, 221, 1079237)-Net in Base 4 — Upper bound on s
There is no (172, 221, 1079238)-net in base 4, because
- 1 times m-reduction [i] would yield (172, 220, 1079238)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 2 839267 100375 187958 692582 812205 705377 788121 942237 466475 729965 593876 544705 498853 459518 771489 480765 234030 757110 605956 376894 584019 954056 > 4220 [i]