Best Known (136, 136+30, s)-Nets in Base 3
(136, 136+30, 698)-Net over F3 — Constructive and digital
Digital (136, 166, 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 (118, 148, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 37, 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, 37, 172)-net over F81, using
- digital (3, 18, 10)-net over F3, using
(136, 136+30, 3634)-Net over F3 — Digital
Digital (136, 166, 3634)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3166, 3634, F3, 30) (dual of [3634, 3468, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3166, 6585, F3, 30) (dual of [6585, 6419, 31]-code), using
- construction XX applied to Ce(30) ⊂ Ce(27) ⊂ Ce(25) [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(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(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(31, 20, F3, 1) (dual of [20, 19, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(31, 4, F3, 1) (dual of [4, 3, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s (see above)
- construction XX applied to Ce(30) ⊂ Ce(27) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(3166, 6585, F3, 30) (dual of [6585, 6419, 31]-code), using
(136, 136+30, 612164)-Net in Base 3 — Upper bound on s
There is no (136, 166, 612165)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 15 926929 481532 286204 739729 487464 539737 672754 288955 047724 605673 453905 235497 813371 > 3166 [i]