Best Known (180, 180+43, s)-Nets in Base 3
(180, 180+43, 696)-Net over F3 — Constructive and digital
Digital (180, 223, 696)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 23, 8)-net over F3, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 2 and N(F) ≥ 8, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- digital (157, 200, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 50, 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, 50, 172)-net over F81, using
- digital (2, 23, 8)-net over F3, using
(180, 180+43, 3054)-Net over F3 — Digital
Digital (180, 223, 3054)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3223, 3054, F3, 43) (dual of [3054, 2831, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(3223, 3288, F3, 43) (dual of [3288, 3065, 44]-code), using
- construction XX applied to Ce(42) ⊂ Ce(40) ⊂ Ce(39) [i] based on
- linear OA(3221, 3281, F3, 43) (dual of [3281, 3060, 44]-code), using an extension Ce(42) of the narrow-sense BCH-code C(I) with length 3280 | 38−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(3217, 3281, F3, 41) (dual of [3281, 3064, 42]-code), using an extension Ce(40) of the narrow-sense BCH-code C(I) with length 3280 | 38−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(3209, 3281, F3, 40) (dual of [3281, 3072, 41]-code), using an extension Ce(39) of the narrow-sense BCH-code C(I) with length 3280 | 38−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(31, 6, F3, 1) (dual of [6, 5, 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(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(42) ⊂ Ce(40) ⊂ Ce(39) [i] based on
- discarding factors / shortening the dual code based on linear OA(3223, 3288, F3, 43) (dual of [3288, 3065, 44]-code), using
(180, 180+43, 480061)-Net in Base 3 — Upper bound on s
There is no (180, 223, 480062)-net in base 3, because
- 1 times m-reduction [i] would yield (180, 222, 480062)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 8335 264241 227151 085728 910877 595041 257069 821720 507075 159537 062281 805157 123139 898578 402937 174665 911111 025525 > 3222 [i]