Best Known (152, 152+34, s)-Nets in Base 3
(152, 152+34, 701)-Net over F3 — Constructive and digital
Digital (152, 186, 701)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (5, 22, 13)-net over F3, using
- net from sequence [i] based on digital (5, 12)-sequence over F3, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F3 with g(F) = 4, N(F) = 12, and 1 place with degree 2 [i] based on function field F/F3 with g(F) = 4 and N(F) ≥ 12, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (5, 12)-sequence over F3, using
- digital (130, 164, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 41, 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, 41, 172)-net over F81, using
- digital (5, 22, 13)-net over F3, using
(152, 152+34, 3636)-Net over F3 — Digital
Digital (152, 186, 3636)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3186, 3636, F3, 34) (dual of [3636, 3450, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3186, 6595, F3, 34) (dual of [6595, 6409, 35]-code), using
- construction XX applied to Ce(33) ⊂ Ce(28) ⊂ Ce(27) [i] based on
- linear OA(3177, 6561, F3, 34) (dual of [6561, 6384, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3153, 6561, F3, 29) (dual of [6561, 6408, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- 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(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(33) ⊂ Ce(28) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3186, 6595, F3, 34) (dual of [6595, 6409, 35]-code), using
(152, 152+34, 595896)-Net in Base 3 — Upper bound on s
There is no (152, 186, 595897)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 55533 650110 001749 516045 457151 470503 394255 871196 732341 129200 054068 506457 614515 552437 140531 > 3186 [i]