Best Known (142, 181, s)-Nets in Base 4
(142, 181, 1052)-Net over F4 — Constructive and digital
Digital (142, 181, 1052)-net over F4, using
- 41 times duplication [i] based on digital (141, 180, 1052)-net over F4, using
- trace code for nets [i] based on digital (6, 45, 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, 45, 263)-net over F256, using
(142, 181, 4118)-Net over F4 — Digital
Digital (142, 181, 4118)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4181, 4118, F4, 39) (dual of [4118, 3937, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(4181, 4121, F4, 39) (dual of [4121, 3940, 40]-code), using
- construction XX applied to Ce(38) ⊂ Ce(34) ⊂ Ce(33) [i] based on
- linear OA(4175, 4096, F4, 39) (dual of [4096, 3921, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(4157, 4096, F4, 35) (dual of [4096, 3939, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(4151, 4096, F4, 34) (dual of [4096, 3945, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(45, 24, F4, 3) (dual of [24, 19, 4]-code or 24-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
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(38) ⊂ Ce(34) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(4181, 4121, F4, 39) (dual of [4121, 3940, 40]-code), using
(142, 181, 1336027)-Net in Base 4 — Upper bound on s
There is no (142, 181, 1336028)-net in base 4, because
- 1 times m-reduction [i] would yield (142, 180, 1336028)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 2 348548 644164 388676 650657 086578 661687 912858 708960 411475 663460 147730 792017 338629 130667 882491 023455 771235 428969 > 4180 [i]