Best Known (108, 108+27, s)-Nets in Base 3
(108, 108+27, 640)-Net over F3 — Constructive and digital
Digital (108, 135, 640)-net over F3, using
- t-expansion [i] based on digital (107, 135, 640)-net over F3, using
- 1 times m-reduction [i] based on digital (107, 136, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 34, 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, 34, 160)-net over F81, using
- 1 times m-reduction [i] based on digital (107, 136, 640)-net over F3, using
(108, 108+27, 1813)-Net over F3 — Digital
Digital (108, 135, 1813)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3135, 1813, F3, 27) (dual of [1813, 1678, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3135, 2216, F3, 27) (dual of [2216, 2081, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,10]) [i] based on
- linear OA(3127, 2188, F3, 27) (dual of [2188, 2061, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(399, 2188, F3, 21) (dual of [2188, 2089, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- construction X applied to C([0,13]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3135, 2216, F3, 27) (dual of [2216, 2081, 28]-code), using
(108, 108+27, 234623)-Net in Base 3 — Upper bound on s
There is no (108, 135, 234624)-net in base 3, because
- 1 times m-reduction [i] would yield (108, 134, 234624)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 8595 122163 063991 648534 757226 221779 503816 240989 063620 450950 996225 > 3134 [i]