Best Known (139, 177, s)-Nets in Base 4
(139, 177, 1052)-Net over F4 — Constructive and digital
Digital (139, 177, 1052)-net over F4, using
- 41 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
(139, 177, 4150)-Net over F4 — Digital
Digital (139, 177, 4150)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4177, 4150, F4, 38) (dual of [4150, 3973, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(4177, 4159, F4, 38) (dual of [4159, 3982, 39]-code), using
- 49 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) [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
- 49 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) [i] based on linear OA(4169, 4102, F4, 38) (dual of [4102, 3933, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(4177, 4159, F4, 38) (dual of [4159, 3982, 39]-code), using
(139, 177, 1073376)-Net in Base 4 — Upper bound on s
There is no (139, 177, 1073377)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 36696 150591 714154 008651 667937 439610 887243 910930 622675 754342 524626 358602 802295 463091 901175 303328 577891 337696 > 4177 [i]