Best Known (83, 103, s)-Nets in Base 3
(83, 103, 640)-Net over F3 — Constructive and digital
Digital (83, 103, 640)-net over F3, using
- 1 times m-reduction [i] based on digital (83, 104, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 26, 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, 26, 160)-net over F81, using
(83, 103, 1892)-Net over F3 — Digital
Digital (83, 103, 1892)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3103, 1892, F3, 20) (dual of [1892, 1789, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3103, 2226, F3, 20) (dual of [2226, 2123, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(13) [i] based on
- linear OA(392, 2187, F3, 20) (dual of [2187, 2095, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(364, 2187, F3, 14) (dual of [2187, 2123, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(19) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(3103, 2226, F3, 20) (dual of [2226, 2123, 21]-code), using
(83, 103, 185897)-Net in Base 3 — Upper bound on s
There is no (83, 103, 185898)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 13 915731 037352 492306 739467 294760 235083 129679 372821 > 3103 [i]