Best Known (164, 164+43, s)-Nets in Base 3
(164, 164+43, 688)-Net over F3 — Constructive and digital
Digital (164, 207, 688)-net over F3, using
- t-expansion [i] based on digital (163, 207, 688)-net over F3, using
- 1 times m-reduction [i] based on digital (163, 208, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 52, 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, 52, 172)-net over F81, using
- 1 times m-reduction [i] based on digital (163, 208, 688)-net over F3, using
(164, 164+43, 1976)-Net over F3 — Digital
Digital (164, 207, 1976)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3207, 1976, F3, 43) (dual of [1976, 1769, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(3207, 2224, F3, 43) (dual of [2224, 2017, 44]-code), using
- construction X applied to C([0,21]) ⊂ C([0,18]) [i] based on
- linear OA(3197, 2188, F3, 43) (dual of [2188, 1991, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,21], and minimum distance d ≥ |{−21,−20,…,21}|+1 = 44 (BCH-bound) [i]
- linear OA(3169, 2188, F3, 37) (dual of [2188, 2019, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (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([0,21]) ⊂ C([0,18]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3207, 2224, F3, 43) (dual of [2224, 2017, 44]-code), using
(164, 164+43, 207850)-Net in Base 3 — Upper bound on s
There is no (164, 207, 207851)-net in base 3, because
- 1 times m-reduction [i] would yield (164, 206, 207851)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 193 633238 380613 787954 086886 888604 962355 535919 844351 244766 244804 957779 184476 381455 682252 580527 140463 > 3206 [i]