Best Known (26−9, 26, s)-Nets in Base 3
(26−9, 26, 60)-Net over F3 — Constructive and digital
Digital (17, 26, 60)-net over F3, using
- trace code for nets [i] based on digital (4, 13, 30)-net over F9, using
- net from sequence [i] based on digital (4, 29)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 4 and N(F) ≥ 30, using
- net from sequence [i] based on digital (4, 29)-sequence over F9, using
(26−9, 26, 80)-Net over F3 — Digital
Digital (17, 26, 80)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(326, 80, F3, 9) (dual of [80, 54, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(326, 91, F3, 9) (dual of [91, 65, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(325, 82, F3, 9) (dual of [82, 57, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 82 | 38−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(317, 82, F3, 7) (dual of [82, 65, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 82 | 38−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(31, 9, F3, 1) (dual of [9, 8, 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
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(326, 91, F3, 9) (dual of [91, 65, 10]-code), using
(26−9, 26, 1058)-Net in Base 3 — Upper bound on s
There is no (17, 26, 1059)-net in base 3, because
- 1 times m-reduction [i] would yield (17, 25, 1059)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 849604 136153 > 325 [i]