Best Known (182, 182+47, s)-Nets in Base 3
(182, 182+47, 688)-Net over F3 — Constructive and digital
Digital (182, 229, 688)-net over F3, using
- t-expansion [i] based on digital (181, 229, 688)-net over F3, using
- 3 times m-reduction [i] based on digital (181, 232, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 58, 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, 58, 172)-net over F81, using
- 3 times m-reduction [i] based on digital (181, 232, 688)-net over F3, using
(182, 182+47, 2255)-Net over F3 — Digital
Digital (182, 229, 2255)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3229, 2255, F3, 47) (dual of [2255, 2026, 48]-code), using
- 50 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 1, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 10 times 0, 1, 15 times 0) [i] based on linear OA(3218, 2194, F3, 47) (dual of [2194, 1976, 48]-code), using
- construction X applied to Ce(46) ⊂ Ce(45) [i] based on
- 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(3211, 2187, F3, 46) (dual of [2187, 1976, 47]-code), using an extension Ce(45) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [i]
- linear OA(30, 7, F3, 0) (dual of [7, 7, 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
- construction X applied to Ce(46) ⊂ Ce(45) [i] based on
- 50 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 1, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 10 times 0, 1, 15 times 0) [i] based on linear OA(3218, 2194, F3, 47) (dual of [2194, 1976, 48]-code), using
(182, 182+47, 252994)-Net in Base 3 — Upper bound on s
There is no (182, 229, 252995)-net in base 3, because
- 1 times m-reduction [i] would yield (182, 228, 252995)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 6 076567 478388 408948 544206 214289 650013 775692 474183 881742 785501 230929 424013 197590 200800 290171 948540 024619 817443 > 3228 [i]