Best Known (222−49, 222, s)-Nets in Base 4
(222−49, 222, 1052)-Net over F4 — Constructive and digital
Digital (173, 222, 1052)-net over F4, using
- 42 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
(222−49, 222, 4112)-Net over F4 — Digital
Digital (173, 222, 4112)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4222, 4112, F4, 49) (dual of [4112, 3890, 50]-code), using
- discarding factors / shortening the dual code based on linear OA(4222, 4119, F4, 49) (dual of [4119, 3897, 50]-code), using
- construction X applied to Ce(48) ⊂ Ce(44) [i] based on
- linear OA(4217, 4096, F4, 49) (dual of [4096, 3879, 50]-code), using an extension Ce(48) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,48], and designed minimum distance d ≥ |I|+1 = 49 [i]
- linear OA(4199, 4096, F4, 45) (dual of [4096, 3897, 46]-code), using an extension Ce(44) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,44], and designed minimum distance d ≥ |I|+1 = 45 [i]
- linear OA(45, 23, F4, 3) (dual of [23, 18, 4]-code or 23-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(48) ⊂ Ce(44) [i] based on
- discarding factors / shortening the dual code based on linear OA(4222, 4119, F4, 49) (dual of [4119, 3897, 50]-code), using
(222−49, 222, 1143413)-Net in Base 4 — Upper bound on s
There is no (173, 222, 1143414)-net in base 4, because
- 1 times m-reduction [i] would yield (173, 221, 1143414)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 11 357070 655420 982678 704989 391352 924180 522674 179002 822832 005752 568155 237230 805953 071561 707023 074993 045043 624067 880258 804821 269762 008764 > 4221 [i]