Best Known (236−39, 236, s)-Nets in Base 3
(236−39, 236, 1480)-Net over F3 — Constructive and digital
Digital (197, 236, 1480)-net over F3, using
- t-expansion [i] based on digital (196, 236, 1480)-net over F3, using
- 4 times m-reduction [i] based on digital (196, 240, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 60, 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, 60, 370)-net over F81, using
- 4 times m-reduction [i] based on digital (196, 240, 1480)-net over F3, using
(236−39, 236, 8981)-Net over F3 — Digital
Digital (197, 236, 8981)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3236, 8981, F3, 2, 39) (dual of [(8981, 2), 17726, 40]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3236, 9851, F3, 2, 39) (dual of [(9851, 2), 19466, 40]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3236, 19702, F3, 39) (dual of [19702, 19466, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3236, 19703, F3, 39) (dual of [19703, 19467, 40]-code), using
- construction X applied to C([0,19]) ⊂ C([0,18]) [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(3217, 19684, F3, 37) (dual of [19684, 19467, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(31, 19, F3, 1) (dual of [19, 18, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,19]) ⊂ C([0,18]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3236, 19703, F3, 39) (dual of [19703, 19467, 40]-code), using
- OOA 2-folding [i] based on linear OA(3236, 19702, F3, 39) (dual of [19702, 19466, 40]-code), using
- discarding factors / shortening the dual code based on linear OOA(3236, 9851, F3, 2, 39) (dual of [(9851, 2), 19466, 40]-NRT-code), using
(236−39, 236, 3158053)-Net in Base 3 — Upper bound on s
There is no (197, 236, 3158054)-net in base 3, because
- 1 times m-reduction [i] would yield (197, 235, 3158054)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 13289 086198 335309 388656 609703 003934 312252 318895 189687 813881 089004 604504 309063 421664 126267 374664 308910 088708 015041 > 3235 [i]