Best Known (201, 201+41, s)-Nets in Base 3
(201, 201+41, 1480)-Net over F3 — Constructive and digital
Digital (201, 242, 1480)-net over F3, using
- t-expansion [i] based on digital (199, 242, 1480)-net over F3, using
- 2 times m-reduction [i] based on digital (199, 244, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 61, 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, 61, 370)-net over F81, using
- 2 times m-reduction [i] based on digital (199, 244, 1480)-net over F3, using
(201, 201+41, 6644)-Net over F3 — Digital
Digital (201, 242, 6644)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3242, 6644, F3, 41) (dual of [6644, 6402, 42]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(3240, 6640, F3, 41) (dual of [6640, 6400, 42]-code), using
- construction X applied to Ce(40) ⊂ Ce(30) [i] based on
- linear OA(3217, 6561, F3, 41) (dual of [6561, 6344, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(3161, 6561, F3, 31) (dual of [6561, 6400, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(323, 79, F3, 9) (dual of [79, 56, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(323, 82, F3, 9) (dual of [82, 59, 10]-code), using
- a “GraX†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(323, 82, F3, 9) (dual of [82, 59, 10]-code), using
- construction X applied to Ce(40) ⊂ Ce(30) [i] based on
- linear OA(3240, 6642, F3, 40) (dual of [6642, 6402, 41]-code), using Gilbert–Varšamov bound and bm = 3240 > Vbs−1(k−1) = 2 821126 306943 998562 425855 832184 150021 675520 147733 841454 224730 354427 592721 082826 374356 418864 891677 434848 897795 326531 [i]
- linear OA(30, 2, F3, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(3240, 6640, F3, 41) (dual of [6640, 6400, 42]-code), using
- construction X with Varšamov bound [i] based on
(201, 201+41, 2331222)-Net in Base 3 — Upper bound on s
There is no (201, 242, 2331223)-net in base 3, because
- 1 times m-reduction [i] would yield (201, 241, 2331223)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 9 687781 504214 536149 487321 773267 568818 088317 455937 937309 217558 461920 846929 185804 585625 702826 060196 534261 853992 836441 > 3241 [i]