Best Known (234−50, 234, s)-Nets in Base 3
(234−50, 234, 688)-Net over F3 — Constructive and digital
Digital (184, 234, 688)-net over F3, using
- 2 times m-reduction [i] based on digital (184, 236, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 59, 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, 59, 172)-net over F81, using
(234−50, 234, 1896)-Net over F3 — Digital
Digital (184, 234, 1896)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3234, 1896, F3, 50) (dual of [1896, 1662, 51]-code), using
- discarding factors / shortening the dual code based on linear OA(3234, 2197, F3, 50) (dual of [2197, 1963, 51]-code), using
- construction XX applied to Ce(49) ⊂ Ce(48) ⊂ Ce(46) [i] based on
- linear OA(3232, 2187, F3, 50) (dual of [2187, 1955, 51]-code), using an extension Ce(49) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,49], and designed minimum distance d ≥ |I|+1 = 50 [i]
- linear OA(3225, 2187, F3, 49) (dual of [2187, 1962, 50]-code), using an extension Ce(48) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,48], and designed minimum distance d ≥ |I|+1 = 49 [i]
- linear OA(3218, 2187, F3, 47) (dual of [2187, 1969, 48]-code), using an extension Ce(46) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,46], and designed minimum distance d ≥ |I|+1 = 47 [i]
- linear OA(30, 8, F3, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(31, 2, F3, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(49) ⊂ Ce(48) ⊂ Ce(46) [i] based on
- discarding factors / shortening the dual code based on linear OA(3234, 2197, F3, 50) (dual of [2197, 1963, 51]-code), using
(234−50, 234, 148723)-Net in Base 3 — Upper bound on s
There is no (184, 234, 148724)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 4430 155425 431879 483168 995749 975264 513727 677862 486163 142698 270901 248208 951745 789507 522823 173915 143860 308044 238569 > 3234 [i]