Best Known (249−45, 249, s)-Nets in Base 3
(249−45, 249, 1480)-Net over F3 — Constructive and digital
Digital (204, 249, 1480)-net over F3, using
- 31 times duplication [i] based on digital (203, 248, 1480)-net over F3, using
- t-expansion [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
- t-expansion [i] based on digital (202, 248, 1480)-net over F3, using
(249−45, 249, 4726)-Net over F3 — Digital
Digital (204, 249, 4726)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3249, 4726, F3, 45) (dual of [4726, 4477, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(3249, 6590, F3, 45) (dual of [6590, 6341, 46]-code), using
- construction X applied to C([0,22]) ⊂ C([0,19]) [i] based on
- linear OA(3241, 6562, F3, 45) (dual of [6562, 6321, 46]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,22], and minimum distance d ≥ |{−22,−21,…,22}|+1 = 46 (BCH-bound) [i]
- 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(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([0,22]) ⊂ C([0,19]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3249, 6590, F3, 45) (dual of [6590, 6341, 46]-code), using
(249−45, 249, 1082111)-Net in Base 3 — Upper bound on s
There is no (204, 249, 1082112)-net in base 3, because
- 1 times m-reduction [i] would yield (204, 248, 1082112)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 21187 115371 077000 045220 696352 705773 893935 468740 115688 509509 812978 406162 800899 433390 839709 957618 464329 219579 852549 165569 > 3248 [i]