Best Known (128, 128+35, s)-Nets in Base 4
(128, 128+35, 1048)-Net over F4 — Constructive and digital
Digital (128, 163, 1048)-net over F4, using
- 1 times m-reduction [i] based on digital (128, 164, 1048)-net over F4, using
- trace code for nets [i] based on digital (5, 41, 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, 41, 262)-net over F256, using
(128, 128+35, 3935)-Net over F4 — Digital
Digital (128, 163, 3935)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4163, 3935, F4, 35) (dual of [3935, 3772, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(4163, 4121, F4, 35) (dual of [4121, 3958, 36]-code), using
- construction XX applied to Ce(34) ⊂ Ce(30) ⊂ Ce(29) [i] based on
- 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(4139, 4096, F4, 31) (dual of [4096, 3957, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(4133, 4096, F4, 30) (dual of [4096, 3963, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [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(34) ⊂ Ce(30) ⊂ Ce(29) [i] based on
- discarding factors / shortening the dual code based on linear OA(4163, 4121, F4, 35) (dual of [4121, 3958, 36]-code), using
(128, 128+35, 1306462)-Net in Base 4 — Upper bound on s
There is no (128, 163, 1306463)-net in base 4, because
- 1 times m-reduction [i] would yield (128, 162, 1306463)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 34 175981 541392 669755 420531 881807 269682 549328 590653 633864 906124 537610 154785 003165 708736 226625 672450 > 4162 [i]