Best Known (173−37, 173, s)-Nets in Base 4
(173−37, 173, 1052)-Net over F4 — Constructive and digital
Digital (136, 173, 1052)-net over F4, using
- 41 times duplication [i] based on digital (135, 172, 1052)-net over F4, using
- trace code for nets [i] based on digital (6, 43, 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, 43, 263)-net over F256, using
(173−37, 173, 4169)-Net over F4 — Digital
Digital (136, 173, 4169)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4173, 4169, F4, 37) (dual of [4169, 3996, 38]-code), using
- 57 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 8 times 0, 1, 13 times 0, 1, 22 times 0) [i] based on linear OA(4164, 4103, F4, 37) (dual of [4103, 3939, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(34) [i] based on
- 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(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(41, 7, F4, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(36) ⊂ Ce(34) [i] based on
- 57 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 8 times 0, 1, 13 times 0, 1, 22 times 0) [i] based on linear OA(4164, 4103, F4, 37) (dual of [4103, 3939, 38]-code), using
(173−37, 173, 1425680)-Net in Base 4 — Upper bound on s
There is no (136, 173, 1425681)-net in base 4, because
- 1 times m-reduction [i] would yield (136, 172, 1425681)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 35 836334 617141 534122 061849 585014 032926 532434 391936 019452 591764 709772 724916 917506 157545 280133 133270 245392 > 4172 [i]