Best Known (187, 226, s)-Nets in Base 3
(187, 226, 1480)-Net over F3 — Constructive and digital
Digital (187, 226, 1480)-net over F3, using
- 2 times m-reduction [i] based on digital (187, 228, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 57, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 57, 370)-net over F81, using
(187, 226, 5803)-Net over F3 — Digital
Digital (187, 226, 5803)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3226, 5803, F3, 39) (dual of [5803, 5577, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3226, 6627, F3, 39) (dual of [6627, 6401, 40]-code), using
- construction X applied to C([0,19]) ⊂ C([0,15]) [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(3161, 6562, F3, 31) (dual of [6562, 6401, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(317, 65, F3, 7) (dual of [65, 48, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(317, 80, F3, 7) (dual of [80, 63, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(317, 80, F3, 7) (dual of [80, 63, 8]-code), using
- construction X applied to C([0,19]) ⊂ C([0,15]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3226, 6627, F3, 39) (dual of [6627, 6401, 40]-code), using
(187, 226, 1771336)-Net in Base 3 — Upper bound on s
There is no (187, 226, 1771337)-net in base 3, because
- 1 times m-reduction [i] would yield (187, 225, 1771337)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 225052 669372 611556 009131 423718 172163 816362 928565 546962 263263 184750 097403 067647 077537 341558 900700 134144 155707 > 3225 [i]