Best Known (169, 169+48, s)-Nets in Base 4
(169, 169+48, 1052)-Net over F4 — Constructive and digital
Digital (169, 217, 1052)-net over F4, using
- 41 times duplication [i] based on digital (168, 216, 1052)-net over F4, using
- trace code for nets [i] based on digital (6, 54, 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, 54, 263)-net over F256, using
(169, 169+48, 3992)-Net over F4 — Digital
Digital (169, 217, 3992)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4217, 3992, F4, 48) (dual of [3992, 3775, 49]-code), using
- discarding factors / shortening the dual code based on linear OA(4217, 4102, F4, 48) (dual of [4102, 3885, 49]-code), using
- 1 times truncation [i] based on linear OA(4218, 4103, F4, 49) (dual of [4103, 3885, 50]-code), using
- construction X applied to Ce(48) ⊂ Ce(46) [i] based on
- linear OA(4217, 4096, F4, 49) (dual of [4096, 3879, 50]-code), using an extension Ce(48) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,48], and designed minimum distance d ≥ |I|+1 = 49 [i]
- 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(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(48) ⊂ Ce(46) [i] based on
- 1 times truncation [i] based on linear OA(4218, 4103, F4, 49) (dual of [4103, 3885, 50]-code), using
- discarding factors / shortening the dual code based on linear OA(4217, 4102, F4, 48) (dual of [4102, 3885, 49]-code), using
(169, 169+48, 907523)-Net in Base 4 — Upper bound on s
There is no (169, 217, 907524)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 44363 239972 888503 218090 311039 565739 245279 067832 088631 008519 814549 926450 248802 825610 914053 006012 615613 937736 651521 552789 385640 913184 > 4217 [i]