Best Known (180−39, 180, s)-Nets in Base 4
(180−39, 180, 1052)-Net over F4 — Constructive and digital
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
(180−39, 180, 3965)-Net over F4 — Digital
Digital (141, 180, 3965)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4180, 3965, F4, 39) (dual of [3965, 3785, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(4180, 4119, F4, 39) (dual of [4119, 3939, 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(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(38) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(4180, 4119, F4, 39) (dual of [4119, 3939, 40]-code), using
(180−39, 180, 1242017)-Net in Base 4 — Upper bound on s
There is no (141, 180, 1242018)-net in base 4, because
- 1 times m-reduction [i] would yield (141, 179, 1242018)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 587139 090620 021617 827013 725522 284147 939784 199096 195480 212177 086597 329152 114737 392175 381294 795874 787557 768112 > 4179 [i]