Best Known (212−45, 212, s)-Nets in Base 3
(212−45, 212, 688)-Net over F3 — Constructive and digital
Digital (167, 212, 688)-net over F3, using
- t-expansion [i] based on digital (166, 212, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 53, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 53, 172)-net over F81, using
(212−45, 212, 1812)-Net over F3 — Digital
Digital (167, 212, 1812)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3212, 1812, F3, 45) (dual of [1812, 1600, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(3212, 2203, F3, 45) (dual of [2203, 1991, 46]-code), using
- construction X applied to C([0,22]) ⊂ C([0,21]) [i] based on
- linear OA(3211, 2188, F3, 45) (dual of [2188, 1977, 46]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,22], and minimum distance d ≥ |{−22,−21,…,22}|+1 = 46 (BCH-bound) [i]
- linear OA(3197, 2188, F3, 43) (dual of [2188, 1991, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,21], and minimum distance d ≥ |{−21,−20,…,21}|+1 = 44 (BCH-bound) [i]
- linear OA(31, 15, F3, 1) (dual of [15, 14, 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([0,22]) ⊂ C([0,21]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3212, 2203, F3, 45) (dual of [2203, 1991, 46]-code), using
(212−45, 212, 170527)-Net in Base 3 — Upper bound on s
There is no (167, 212, 170528)-net in base 3, because
- 1 times m-reduction [i] would yield (167, 211, 170528)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 47054 716392 543959 518232 543752 223469 064305 802115 031186 856886 108904 542326 258325 260781 893280 240093 912257 > 3211 [i]