Best Known (191, 191+43, s)-Nets in Base 3
(191, 191+43, 896)-Net over F3 — Constructive and digital
Digital (191, 234, 896)-net over F3, using
- t-expansion [i] based on digital (190, 234, 896)-net over F3, using
- 2 times m-reduction [i] based on digital (190, 236, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 59, 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, 59, 224)-net over F81, using
- 2 times m-reduction [i] based on digital (190, 236, 896)-net over F3, using
(191, 191+43, 4114)-Net over F3 — Digital
Digital (191, 234, 4114)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3234, 4114, F3, 43) (dual of [4114, 3880, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(3234, 6595, F3, 43) (dual of [6595, 6361, 44]-code), using
- construction XX applied to Ce(42) ⊂ Ce(37) ⊂ Ce(36) [i] based on
- linear OA(3225, 6561, F3, 43) (dual of [6561, 6336, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(3201, 6561, F3, 38) (dual of [6561, 6360, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- 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(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(42) ⊂ Ce(37) ⊂ Ce(36) [i] based on
- discarding factors / shortening the dual code based on linear OA(3234, 6595, F3, 43) (dual of [6595, 6361, 44]-code), using
(191, 191+43, 853544)-Net in Base 3 — Upper bound on s
There is no (191, 234, 853545)-net in base 3, because
- 1 times m-reduction [i] would yield (191, 233, 853545)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1476 589960 988075 319763 667070 315536 894306 580736 421093 645727 071385 833445 944572 608882 105652 756281 345963 476291 268483 > 3233 [i]