Best Known (138−28, 138, s)-Nets in Base 3
(138−28, 138, 640)-Net over F3 — Constructive and digital
Digital (110, 138, 640)-net over F3, using
- 2 times m-reduction [i] based on digital (110, 140, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 35, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- trace code for nets [i] based on digital (5, 35, 160)-net over F81, using
(138−28, 138, 1699)-Net over F3 — Digital
Digital (110, 138, 1699)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3138, 1699, F3, 28) (dual of [1699, 1561, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3138, 2226, F3, 28) (dual of [2226, 2088, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(21) [i] based on
- linear OA(3127, 2187, F3, 28) (dual of [2187, 2060, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(399, 2187, F3, 22) (dual of [2187, 2088, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(311, 39, F3, 5) (dual of [39, 28, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 40, F3, 5) (dual of [40, 29, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(31, 13, F3, 1) (dual of [13, 12, 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, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- linear OA(37, 14, F3, 5) (dual of [14, 7, 6]-code), using
- extended quadratic residue code Qe(14,3) [i]
- linear OA(31, 13, F3, 1) (dual of [13, 12, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(311, 40, F3, 5) (dual of [40, 29, 6]-code), using
- construction X applied to Ce(27) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3138, 2226, F3, 28) (dual of [2226, 2088, 29]-code), using
(138−28, 138, 152560)-Net in Base 3 — Upper bound on s
There is no (110, 138, 152561)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 696239 955921 931079 939527 157268 840945 865762 591638 704306 125720 521177 > 3138 [i]