Best Known (175, 175+32, s)-Nets in Base 3
(175, 175+32, 1480)-Net over F3 — Constructive and digital
Digital (175, 207, 1480)-net over F3, using
- 5 times m-reduction [i] based on digital (175, 212, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 53, 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, 53, 370)-net over F81, using
(175, 175+32, 11347)-Net over F3 — Digital
Digital (175, 207, 11347)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3207, 11347, F3, 32) (dual of [11347, 11140, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3207, 19725, F3, 32) (dual of [19725, 19518, 33]-code), using
- construction X applied to C([0,16]) ⊂ C([0,13]) [i] based on
- 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(3163, 19684, F3, 27) (dual of [19684, 19521, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- construction X applied to C([0,16]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3207, 19725, F3, 32) (dual of [19725, 19518, 33]-code), using
(175, 175+32, 5061318)-Net in Base 3 — Upper bound on s
There is no (175, 207, 5061319)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 580 899475 506102 354305 935539 421346 641270 159109 941226 054024 219062 160073 532033 352968 508393 069720 002785 > 3207 [i]