Best Known (125, 125+34, s)-Nets in Base 4
(125, 125+34, 1048)-Net over F4 — Constructive and digital
Digital (125, 159, 1048)-net over F4, using
- 1 times m-reduction [i] based on digital (125, 160, 1048)-net over F4, using
- trace code for nets [i] based on digital (5, 40, 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, 40, 262)-net over F256, using
(125, 125+34, 3978)-Net over F4 — Digital
Digital (125, 159, 3978)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4159, 3978, F4, 34) (dual of [3978, 3819, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(4159, 4128, F4, 34) (dual of [4128, 3969, 35]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4158, 4127, F4, 34) (dual of [4127, 3969, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(28) [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(4127, 4096, F4, 29) (dual of [4096, 3969, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(47, 31, F4, 4) (dual of [31, 24, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(33) ⊂ Ce(28) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4158, 4127, F4, 34) (dual of [4127, 3969, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(4159, 4128, F4, 34) (dual of [4128, 3969, 35]-code), using
(125, 125+34, 1022939)-Net in Base 4 — Upper bound on s
There is no (125, 159, 1022940)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 534003 608481 865282 014192 152143 787117 552092 594990 212205 736306 219492 225350 156256 152493 303432 098975 > 4159 [i]