Best Known (238−53, 238, s)-Nets in Base 4
(238−53, 238, 1052)-Net over F4 — Constructive and digital
Digital (185, 238, 1052)-net over F4, using
- 42 times duplication [i] based on digital (183, 236, 1052)-net over F4, using
- trace code for nets [i] based on digital (6, 59, 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, 59, 263)-net over F256, using
(238−53, 238, 4111)-Net over F4 — Digital
Digital (185, 238, 4111)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4238, 4111, F4, 53) (dual of [4111, 3873, 54]-code), using
- construction X applied to Ce(52) ⊂ Ce(49) [i] based on
- linear OA(4235, 4096, F4, 53) (dual of [4096, 3861, 54]-code), using an extension Ce(52) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,52], and designed minimum distance d ≥ |I|+1 = 53 [i]
- linear OA(4223, 4096, F4, 50) (dual of [4096, 3873, 51]-code), using an extension Ce(49) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,49], and designed minimum distance d ≥ |I|+1 = 50 [i]
- linear OA(43, 15, F4, 2) (dual of [15, 12, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- construction X applied to Ce(52) ⊂ Ce(49) [i] based on
(238−53, 238, 1081861)-Net in Base 4 — Upper bound on s
There is no (185, 238, 1081862)-net in base 4, because
- 1 times m-reduction [i] would yield (185, 237, 1081862)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 48777 894055 607090 872184 292349 961501 338676 430858 867088 057861 821860 324701 349696 627906 111717 903108 061604 933380 009170 223410 533976 200521 743693 218432 > 4237 [i]