Best Known (246−42, 246, s)-Nets in Base 3
(246−42, 246, 1480)-Net over F3 — Constructive and digital
Digital (204, 246, 1480)-net over F3, using
- t-expansion [i] based on digital (202, 246, 1480)-net over F3, using
- 2 times m-reduction [i] based on digital (202, 248, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 62, 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, 62, 370)-net over F81, using
- 2 times m-reduction [i] based on digital (202, 248, 1480)-net over F3, using
(246−42, 246, 6555)-Net over F3 — Digital
Digital (204, 246, 6555)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3246, 6555, F3, 42) (dual of [6555, 6309, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(3246, 6631, F3, 42) (dual of [6631, 6385, 43]-code), using
- construction X applied to C([0,21]) ⊂ C([0,16]) [i] based on
- linear OA(3225, 6562, F3, 43) (dual of [6562, 6337, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,21], and minimum distance d ≥ |{−21,−20,…,21}|+1 = 44 (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(321, 69, F3, 8) (dual of [69, 48, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(321, 80, F3, 8) (dual of [80, 59, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(321, 80, F3, 8) (dual of [80, 59, 9]-code), using
- construction X applied to C([0,21]) ⊂ C([0,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3246, 6631, F3, 42) (dual of [6631, 6385, 43]-code), using
(246−42, 246, 1684969)-Net in Base 3 — Upper bound on s
There is no (204, 246, 1684970)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 2354 141625 656887 703071 970241 711039 678789 693189 891139 671239 840500 921164 559291 004427 417460 827203 077825 058195 929434 366093 > 3246 [i]