Best Known (171−30, 171, s)-Nets in Base 3
(171−30, 171, 704)-Net over F3 — Constructive and digital
Digital (141, 171, 704)-net over F3, using
- 31 times duplication [i] based on digital (140, 170, 704)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (7, 22, 16)-net over F3, using
- net from sequence [i] based on digital (7, 15)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 7 and N(F) ≥ 16, using
- net from sequence [i] based on digital (7, 15)-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 (7, 22, 16)-net over F3, using
- (u, u+v)-construction [i] based on
(171−30, 171, 4428)-Net over F3 — Digital
Digital (141, 171, 4428)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3171, 4428, F3, 30) (dual of [4428, 4257, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3171, 6604, F3, 30) (dual of [6604, 6433, 31]-code), using
- 1 times truncation [i] based on linear OA(3172, 6605, F3, 31) (dual of [6605, 6433, 32]-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(311, 43, F3, 5) (dual of [43, 32, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- 1 times truncation [i] based on linear OA(3172, 6605, F3, 31) (dual of [6605, 6433, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3171, 6604, F3, 30) (dual of [6604, 6433, 31]-code), using
(171−30, 171, 882900)-Net in Base 3 — Upper bound on s
There is no (141, 171, 882901)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 3870 235926 527434 241647 107800 590642 829776 034474 523550 556684 383524 220560 876326 209467 > 3171 [i]