Best Known (127, 127+31, s)-Nets in Base 3
(127, 127+31, 688)-Net over F3 — Constructive and digital
Digital (127, 158, 688)-net over F3, using
- 2 times m-reduction [i] based on digital (127, 160, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 40, 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, 40, 172)-net over F81, using
(127, 127+31, 2207)-Net over F3 — Digital
Digital (127, 158, 2207)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3158, 2207, F3, 31) (dual of [2207, 2049, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3158, 2239, F3, 31) (dual of [2239, 2081, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(22) [i] based on
- linear OA(3141, 2187, F3, 31) (dual of [2187, 2046, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3106, 2187, F3, 23) (dual of [2187, 2081, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(317, 52, F3, 7) (dual of [52, 35, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(317, 80, F3, 7) (dual of [80, 63, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(317, 80, F3, 7) (dual of [80, 63, 8]-code), using
- construction X applied to Ce(30) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(3158, 2239, F3, 31) (dual of [2239, 2081, 32]-code), using
(127, 127+31, 316654)-Net in Base 3 — Upper bound on s
There is no (127, 158, 316655)-net in base 3, because
- 1 times m-reduction [i] would yield (127, 157, 316655)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 809 177621 133471 475983 737802 181474 513519 031412 427638 553793 690396 287286 410147 > 3157 [i]