Best Known (169−30, 169, s)-Nets in Base 3
(169−30, 169, 702)-Net over F3 — Constructive and digital
Digital (139, 169, 702)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (6, 21, 14)-net over F3, using
- net from sequence [i] based on digital (6, 13)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 6 and N(F) ≥ 14, using
- net from sequence [i] based on digital (6, 13)-sequence over F3, using
- digital (118, 148, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 37, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 37, 172)-net over F81, using
- digital (6, 21, 14)-net over F3, using
(169−30, 169, 4092)-Net over F3 — Digital
Digital (139, 169, 4092)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3169, 4092, F3, 30) (dual of [4092, 3923, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3169, 6602, F3, 30) (dual of [6602, 6433, 31]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- linear OA(3161, 6562, F3, 31) (dual of [6562, 6401, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(3129, 6562, F3, 25) (dual of [6562, 6433, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(38, 40, F3, 4) (dual of [40, 32, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3169, 6602, F3, 30) (dual of [6602, 6433, 31]-code), using
(169−30, 169, 762596)-Net in Base 3 — Upper bound on s
There is no (139, 169, 762597)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 430 031801 518214 576759 535204 595327 626334 626220 996335 367530 276064 593421 993779 178491 > 3169 [i]