Best Known (163, 209, s)-Nets in Base 4
(163, 209, 1052)-Net over F4 — Constructive and digital
Digital (163, 209, 1052)-net over F4, using
- 41 times duplication [i] based on digital (162, 208, 1052)-net over F4, using
- trace code for nets [i] based on digital (6, 52, 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, 52, 263)-net over F256, using
(163, 209, 4001)-Net over F4 — Digital
Digital (163, 209, 4001)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4209, 4001, F4, 46) (dual of [4001, 3792, 47]-code), using
- discarding factors / shortening the dual code based on linear OA(4209, 4113, F4, 46) (dual of [4113, 3904, 47]-code), using
- construction X applied to Ce(45) ⊂ Ce(41) [i] based on
- 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(4187, 4096, F4, 42) (dual of [4096, 3909, 43]-code), using an extension Ce(41) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,41], and designed minimum distance d ≥ |I|+1 = 42 [i]
- linear OA(44, 17, F4, 3) (dual of [17, 13, 4]-code or 17-cap in PG(3,4)), using
- construction X applied to Ce(45) ⊂ Ce(41) [i] based on
- discarding factors / shortening the dual code based on linear OA(4209, 4113, F4, 46) (dual of [4113, 3904, 47]-code), using
(163, 209, 929435)-Net in Base 4 — Upper bound on s
There is no (163, 209, 929436)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 676929 298317 063656 765072 197682 953669 143819 223221 716558 926696 771776 557958 998155 791462 012506 806008 571876 922180 790787 756084 547554 > 4209 [i]