Best Known (202−37, 202, s)-Nets in Base 3
(202−37, 202, 896)-Net over F3 — Constructive and digital
Digital (165, 202, 896)-net over F3, using
- 32 times duplication [i] based on digital (163, 200, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 50, 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, 50, 224)-net over F81, using
(202−37, 202, 3789)-Net over F3 — Digital
Digital (165, 202, 3789)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3202, 3789, F3, 37) (dual of [3789, 3587, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3202, 6595, F3, 37) (dual of [6595, 6393, 38]-code), using
- construction XX applied to Ce(36) ⊂ Ce(31) ⊂ Ce(30) [i] based on
- linear OA(3193, 6561, F3, 37) (dual of [6561, 6368, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(3169, 6561, F3, 32) (dual of [6561, 6392, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [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(38, 33, F3, 4) (dual of [33, 25, 5]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- linear OA(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(36) ⊂ Ce(31) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(3202, 6595, F3, 37) (dual of [6595, 6393, 38]-code), using
(202−37, 202, 803423)-Net in Base 3 — Upper bound on s
There is no (165, 202, 803424)-net in base 3, because
- 1 times m-reduction [i] would yield (165, 201, 803424)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 796848 187482 306405 275117 907452 792131 422409 130357 055151 089667 093622 973131 890189 538798 311804 480065 > 3201 [i]