Best Known (178, 217, s)-Nets in Base 3
(178, 217, 896)-Net over F3 — Constructive and digital
Digital (178, 217, 896)-net over F3, using
- 3 times m-reduction [i] based on digital (178, 220, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 55, 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, 55, 224)-net over F81, using
(178, 217, 4434)-Net over F3 — Digital
Digital (178, 217, 4434)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3217, 4434, F3, 39) (dual of [4434, 4217, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3217, 6590, F3, 39) (dual of [6590, 6373, 40]-code), using
- construction X applied to C([0,19]) ⊂ C([0,16]) [i] based on
- linear OA(3209, 6562, F3, 39) (dual of [6562, 6353, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,19], and minimum distance d ≥ |{−19,−18,…,19}|+1 = 40 (BCH-bound) [i]
- linear OA(3177, 6562, F3, 33) (dual of [6562, 6385, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (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([0,19]) ⊂ C([0,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3217, 6590, F3, 39) (dual of [6590, 6373, 40]-code), using
(178, 217, 1052672)-Net in Base 3 — Upper bound on s
There is no (178, 217, 1052673)-net in base 3, because
- 1 times m-reduction [i] would yield (178, 216, 1052673)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 11 433955 655037 344929 840498 129303 115155 026754 057548 589076 540710 691186 313409 716205 884637 882645 470489 506779 > 3216 [i]