Best Known (89, 110, s)-Nets in Base 3
(89, 110, 640)-Net over F3 — Constructive and digital
Digital (89, 110, 640)-net over F3, using
- 2 times m-reduction [i] based on digital (89, 112, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 28, 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, 28, 160)-net over F81, using
(89, 110, 2147)-Net over F3 — Digital
Digital (89, 110, 2147)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3110, 2147, F3, 21) (dual of [2147, 2037, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3110, 2227, F3, 21) (dual of [2227, 2117, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- 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(371, 2188, F3, 15) (dual of [2188, 2117, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [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 C([0,10]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3110, 2227, F3, 21) (dual of [2227, 2117, 22]-code), using
(89, 110, 359382)-Net in Base 3 — Upper bound on s
There is no (89, 110, 359383)-net in base 3, because
- 1 times m-reduction [i] would yield (89, 109, 359383)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 10144 415295 783673 363510 379415 854846 329455 671735 802141 > 3109 [i]