Best Known (183, 226, s)-Nets in Base 3
(183, 226, 896)-Net over F3 — Constructive and digital
Digital (183, 226, 896)-net over F3, using
- 32 times duplication [i] based on digital (181, 224, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 56, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- trace code for nets [i] based on digital (13, 56, 224)-net over F81, using
(183, 226, 3313)-Net over F3 — Digital
Digital (183, 226, 3313)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3226, 3313, F3, 43) (dual of [3313, 3087, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(3226, 6570, F3, 43) (dual of [6570, 6344, 44]-code), using
- construction X applied to Ce(42) ⊂ Ce(40) [i] based on
- linear OA(3225, 6561, F3, 43) (dual of [6561, 6336, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(3217, 6561, F3, 41) (dual of [6561, 6344, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [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 Ce(42) ⊂ Ce(40) [i] based on
- discarding factors / shortening the dual code based on linear OA(3226, 6570, F3, 43) (dual of [6570, 6344, 44]-code), using
(183, 226, 561642)-Net in Base 3 — Upper bound on s
There is no (183, 226, 561643)-net in base 3, because
- 1 times m-reduction [i] would yield (183, 225, 561643)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 225053 739149 927934 680411 360074 672464 754329 098962 354351 804693 154764 195947 397231 825328 223327 843591 238288 550511 > 3225 [i]