Best Known (140, 140+38, s)-Nets in Base 4
(140, 140+38, 1052)-Net over F4 — Constructive and digital
Digital (140, 178, 1052)-net over F4, using
- 42 times duplication [i] based on digital (138, 176, 1052)-net over F4, using
- trace code for nets [i] based on digital (6, 44, 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, 44, 263)-net over F256, using
(140, 140+38, 4192)-Net over F4 — Digital
Digital (140, 178, 4192)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4178, 4192, F4, 38) (dual of [4192, 4014, 39]-code), using
- 81 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 4 times 0, 1, 7 times 0, 1, 12 times 0, 1, 19 times 0, 1, 31 times 0) [i] based on linear OA(4169, 4102, F4, 38) (dual of [4102, 3933, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(36) [i] based on
- linear OA(4169, 4096, F4, 38) (dual of [4096, 3927, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(4163, 4096, F4, 37) (dual of [4096, 3933, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [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(37) ⊂ Ce(36) [i] based on
- 81 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 4 times 0, 1, 7 times 0, 1, 12 times 0, 1, 19 times 0, 1, 31 times 0) [i] based on linear OA(4169, 4102, F4, 38) (dual of [4102, 3933, 39]-code), using
(140, 140+38, 1154622)-Net in Base 4 — Upper bound on s
There is no (140, 178, 1154623)-net in base 4, because
- 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]