Best Known (162−34, 162, s)-Nets in Base 4
(162−34, 162, 1052)-Net over F4 — Constructive and digital
Digital (128, 162, 1052)-net over F4, using
- 42 times duplication [i] based on digital (126, 160, 1052)-net over F4, using
- trace code for nets [i] based on digital (6, 40, 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, 40, 263)-net over F256, using
(162−34, 162, 4252)-Net over F4 — Digital
Digital (128, 162, 4252)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4162, 4252, F4, 34) (dual of [4252, 4090, 35]-code), using
- 139 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 13 times 0, 1, 20 times 0, 1, 33 times 0, 1, 50 times 0) [i] based on linear OA(4151, 4102, F4, 34) (dual of [4102, 3951, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(32) [i] based on
- 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(4145, 4096, F4, 33) (dual of [4096, 3951, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(40, 6, F4, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(33) ⊂ Ce(32) [i] based on
- 139 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 13 times 0, 1, 20 times 0, 1, 33 times 0, 1, 50 times 0) [i] based on linear OA(4151, 4102, F4, 34) (dual of [4102, 3951, 35]-code), using
(162−34, 162, 1306462)-Net in Base 4 — Upper bound on s
There is no (128, 162, 1306463)-net in base 4, because
- 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]