Best Known (126, 126+28, s)-Nets in Base 3
(126, 126+28, 692)-Net over F3 — Constructive and digital
Digital (126, 154, 692)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (0, 14, 4)-net over F3, using
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 0 and N(F) ≥ 4, using
- the rational function field F3(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- digital (112, 140, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 35, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 35, 172)-net over F81, using
- digital (0, 14, 4)-net over F3, using
(126, 126+28, 3363)-Net over F3 — Digital
Digital (126, 154, 3363)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3154, 3363, F3, 28) (dual of [3363, 3209, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3154, 6595, F3, 28) (dual of [6595, 6441, 29]-code), using
- construction XX applied to Ce(27) ⊂ Ce(22) ⊂ Ce(21) [i] based on
- linear OA(3145, 6561, F3, 28) (dual of [6561, 6416, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3121, 6561, F3, 23) (dual of [6561, 6440, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3113, 6561, F3, 22) (dual of [6561, 6448, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(27) ⊂ Ce(22) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3154, 6595, F3, 28) (dual of [6595, 6441, 29]-code), using
(126, 126+28, 535489)-Net in Base 3 — Upper bound on s
There is no (126, 154, 535490)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 29 969809 140264 459174 194494 812890 690213 518728 654252 154395 183130 510530 285837 > 3154 [i]