Best Known (171−31, 171, s)-Nets in Base 3
(171−31, 171, 700)-Net over F3 — Constructive and digital
Digital (140, 171, 700)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (4, 19, 12)-net over F3, using
- net from sequence [i] based on digital (4, 11)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 4 and N(F) ≥ 12, using
- net from sequence [i] based on digital (4, 11)-sequence over F3, using
- digital (121, 152, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 38, 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, 38, 172)-net over F81, using
- digital (4, 19, 12)-net over F3, using
(171−31, 171, 3629)-Net over F3 — Digital
Digital (140, 171, 3629)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3171, 3629, F3, 31) (dual of [3629, 3458, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3171, 6598, F3, 31) (dual of [6598, 6427, 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(310, 36, F3, 5) (dual of [36, 26, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(31, 12, F3, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(33, 12, F3, 2) (dual of [12, 9, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- linear OA(36, 12, F3, 5) (dual of [12, 6, 6]-code), using
- extended Golay code Ge(3) [i]
- linear OA(31, 12, F3, 1) (dual of [12, 11, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3171, 6598, F3, 31) (dual of [6598, 6427, 32]-code), using
(171−31, 171, 820546)-Net in Base 3 — Upper bound on s
There is no (140, 171, 820547)-net in base 3, because
- 1 times m-reduction [i] would yield (140, 170, 820547)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1290 082158 455814 533487 639331 666852 855973 578079 359958 713655 481096 956079 300525 325811 > 3170 [i]