Best Known (203, 203+47, s)-Nets in Base 3
(203, 203+47, 896)-Net over F3 — Constructive and digital
Digital (203, 250, 896)-net over F3, using
- 32 times duplication [i] based on digital (201, 248, 896)-net over F3, using
- t-expansion [i] based on digital (199, 248, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 62, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- trace code for nets [i] based on digital (13, 62, 224)-net over F81, using
- t-expansion [i] based on digital (199, 248, 896)-net over F3, using
(203, 203+47, 3805)-Net over F3 — Digital
Digital (203, 250, 3805)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3250, 3805, F3, 47) (dual of [3805, 3555, 48]-code), using
- discarding factors / shortening the dual code based on linear OA(3250, 6570, F3, 47) (dual of [6570, 6320, 48]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3249, 6569, F3, 47) (dual of [6569, 6320, 48]-code), using
- construction X applied to Ce(46) ⊂ Ce(45) [i] based on
- linear OA(3249, 6561, F3, 47) (dual of [6561, 6312, 48]-code), using an extension Ce(46) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,46], and designed minimum distance d ≥ |I|+1 = 47 [i]
- linear OA(3241, 6561, F3, 46) (dual of [6561, 6320, 47]-code), using an extension Ce(45) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [i]
- linear OA(30, 8, F3, 0) (dual of [8, 8, 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
- construction X applied to Ce(46) ⊂ Ce(45) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3249, 6569, F3, 47) (dual of [6569, 6320, 48]-code), using
- discarding factors / shortening the dual code based on linear OA(3250, 6570, F3, 47) (dual of [6570, 6320, 48]-code), using
(203, 203+47, 689871)-Net in Base 3 — Upper bound on s
There is no (203, 250, 689872)-net in base 3, because
- 1 times m-reduction [i] would yield (203, 249, 689872)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 63561 943602 899053 188333 576385 023620 608626 441234 155967 641694 706603 170914 074146 205257 916937 395624 265054 362320 808096 633281 > 3249 [i]