Best Known (139, 139+31, s)-Nets in Base 3
(139, 139+31, 698)-Net over F3 — Constructive and digital
Digital (139, 170, 698)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (3, 18, 10)-net over F3, using
- net from sequence [i] based on digital (3, 9)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 3 and N(F) ≥ 10, using
- net from sequence [i] based on digital (3, 9)-sequence over F3, using
- digital (121, 152, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 38, 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, 38, 172)-net over F81, using
- digital (3, 18, 10)-net over F3, using
(139, 139+31, 3493)-Net over F3 — Digital
Digital (139, 170, 3493)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3170, 3493, F3, 31) (dual of [3493, 3323, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3170, 6595, F3, 31) (dual of [6595, 6425, 32]-code), using
- construction XX applied to Ce(30) ⊂ Ce(25) ⊂ Ce(24) [i] based on
- 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(3137, 6561, F3, 26) (dual of [6561, 6424, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(3129, 6561, F3, 25) (dual of [6561, 6432, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(30) ⊂ Ce(25) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(3170, 6595, F3, 31) (dual of [6595, 6425, 32]-code), using
(139, 139+31, 762596)-Net in Base 3 — Upper bound on s
There is no (139, 170, 762597)-net in base 3, because
- 1 times m-reduction [i] would yield (139, 169, 762597)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 430 031801 518214 576759 535204 595327 626334 626220 996335 367530 276064 593421 993779 178491 > 3169 [i]