Best Known (167, 214, s)-Nets in Base 4
(167, 214, 1052)-Net over F4 — Constructive and digital
Digital (167, 214, 1052)-net over F4, using
- 42 times duplication [i] based on digital (165, 212, 1052)-net over F4, using
- trace code for nets [i] based on digital (6, 53, 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, 53, 263)-net over F256, using
(167, 214, 4117)-Net over F4 — Digital
Digital (167, 214, 4117)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4214, 4117, F4, 47) (dual of [4117, 3903, 48]-code), using
- 12 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 8 times 0) [i] based on linear OA(4211, 4102, F4, 47) (dual of [4102, 3891, 48]-code), using
- construction X applied to Ce(46) ⊂ Ce(45) [i] based on
- linear OA(4211, 4096, F4, 47) (dual of [4096, 3885, 48]-code), using an extension Ce(46) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,46], and designed minimum distance d ≥ |I|+1 = 47 [i]
- linear OA(4205, 4096, F4, 46) (dual of [4096, 3891, 47]-code), using an extension Ce(45) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [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(46) ⊂ Ce(45) [i] based on
- 12 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 8 times 0) [i] based on linear OA(4211, 4102, F4, 47) (dual of [4102, 3891, 48]-code), using
(167, 214, 1182843)-Net in Base 4 — Upper bound on s
There is no (167, 214, 1182844)-net in base 4, because
- 1 times m-reduction [i] would yield (167, 213, 1182844)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 173 294149 597920 352475 235320 965980 445675 756644 209017 279568 952583 293659 031333 122775 910688 618568 689745 035837 341005 470406 770735 314558 > 4213 [i]