Best Known (230−39, 230, s)-Nets in Base 3
(230−39, 230, 1480)-Net over F3 — Constructive and digital
Digital (191, 230, 1480)-net over F3, using
- t-expansion [i] based on digital (190, 230, 1480)-net over F3, using
- 2 times m-reduction [i] based on digital (190, 232, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 58, 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, 58, 370)-net over F81, using
- 2 times m-reduction [i] based on digital (190, 232, 1480)-net over F3, using
(230−39, 230, 6539)-Net over F3 — Digital
Digital (191, 230, 6539)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3230, 6539, F3, 39) (dual of [6539, 6309, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3230, 6631, F3, 39) (dual of [6631, 6401, 40]-code), using
- 5 times code embedding in larger space [i] based on linear OA(3225, 6626, F3, 39) (dual of [6626, 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(316, 64, F3, 7) (dual of [64, 48, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(316, 82, F3, 7) (dual of [82, 66, 8]-code), using
- construction X applied to C([0,19]) ⊂ C([0,15]) [i] based on
- 5 times code embedding in larger space [i] based on linear OA(3225, 6626, F3, 39) (dual of [6626, 6401, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3230, 6631, F3, 39) (dual of [6631, 6401, 40]-code), using
(230−39, 230, 2232279)-Net in Base 3 — Upper bound on s
There is no (191, 230, 2232280)-net in base 3, because
- 1 times m-reduction [i] would yield (191, 229, 2232280)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 18 229232 218811 717777 425548 011262 327386 529700 464452 391436 186032 679263 269358 748198 052745 689012 448352 332601 562337 > 3229 [i]