Best Known (137, 172, s)-Nets in Base 3
(137, 172, 688)-Net over F3 — Constructive and digital
Digital (137, 172, 688)-net over F3, using
- t-expansion [i] based on digital (136, 172, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 43, 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, 43, 172)-net over F81, using
(137, 172, 1922)-Net over F3 — Digital
Digital (137, 172, 1922)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3172, 1922, F3, 35) (dual of [1922, 1750, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3172, 2223, F3, 35) (dual of [2223, 2051, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(28) [i] based on
- linear OA(3162, 2187, F3, 35) (dual of [2187, 2025, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(3134, 2187, F3, 29) (dual of [2187, 2053, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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 Ce(34) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(3172, 2223, F3, 35) (dual of [2223, 2051, 36]-code), using
(137, 172, 226027)-Net in Base 3 — Upper bound on s
There is no (137, 172, 226028)-net in base 3, because
- 1 times m-reduction [i] would yield (137, 171, 226028)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 3870 244931 716037 874056 131658 404812 870544 660210 550300 687785 924111 634088 705464 414169 > 3171 [i]