Best Known (140, 140+39, s)-Nets in Base 4
(140, 140+39, 1048)-Net over F4 — Constructive and digital
Digital (140, 179, 1048)-net over F4, using
- 1 times m-reduction [i] based on digital (140, 180, 1048)-net over F4, using
- trace code for nets [i] based on digital (5, 45, 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, 45, 262)-net over F256, using
(140, 140+39, 3818)-Net over F4 — Digital
Digital (140, 179, 3818)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4179, 3818, F4, 39) (dual of [3818, 3639, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(4179, 4113, F4, 39) (dual of [4113, 3934, 40]-code), using
- construction X applied to Ce(38) ⊂ Ce(34) [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(44, 17, F4, 3) (dual of [17, 13, 4]-code or 17-cap in PG(3,4)), using
- construction X applied to Ce(38) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(4179, 4113, F4, 39) (dual of [4113, 3934, 40]-code), using
(140, 140+39, 1154622)-Net in Base 4 — Upper bound on s
There is no (140, 179, 1154623)-net in base 4, because
- 1 times m-reduction [i] would yield (140, 178, 1154623)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 146785 339204 635636 380003 613759 619609 252924 457353 742528 936341 699619 828893 406619 819920 665707 061941 136913 454376 > 4178 [i]