Best Known (207, 207+39, s)-Nets in Base 3
(207, 207+39, 1496)-Net over F3 — Constructive and digital
Digital (207, 246, 1496)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (7, 26, 16)-net over F3, using
- net from sequence [i] based on digital (7, 15)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 7 and N(F) ≥ 16, using
- net from sequence [i] based on digital (7, 15)-sequence over F3, using
- digital (181, 220, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 55, 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, 55, 370)-net over F81, using
- digital (7, 26, 16)-net over F3, using
(207, 207+39, 10537)-Net over F3 — Digital
Digital (207, 246, 10537)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3246, 10537, F3, 39) (dual of [10537, 10291, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3246, 19725, F3, 39) (dual of [19725, 19479, 40]-code), using
- construction X applied to C([0,19]) ⊂ C([0,16]) [i] based on
- linear OA(3235, 19684, F3, 39) (dual of [19684, 19449, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,19], and minimum distance d ≥ |{−19,−18,…,19}|+1 = 40 (BCH-bound) [i]
- linear OA(3199, 19684, F3, 33) (dual of [19684, 19485, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(311, 41, F3, 5) (dual of [41, 30, 6]-code), using
- (u, u+v)-construction [i] based on
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [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
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- (u, u+v)-construction [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(3246, 19725, F3, 39) (dual of [19725, 19479, 40]-code), using
(207, 207+39, 5630372)-Net in Base 3 — Upper bound on s
There is no (207, 246, 5630373)-net in base 3, because
- 1 times m-reduction [i] would yield (207, 245, 5630373)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 784 708902 269655 978308 436116 173962 673375 393807 577645 614108 407693 762596 932104 825145 027963 185775 392261 911470 357148 057259 > 3245 [i]