Best Known (103, 103+21, s)-Nets in Base 3
(103, 103+21, 696)-Net over F3 — Constructive and digital
Digital (103, 124, 696)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 12, 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 (91, 112, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 28, 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, 28, 172)-net over F81, using
- digital (2, 12, 8)-net over F3, using
(103, 103+21, 4845)-Net over F3 — Digital
Digital (103, 124, 4845)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3124, 4845, F3, 21) (dual of [4845, 4721, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3124, 6603, F3, 21) (dual of [6603, 6479, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- linear OA(3113, 6562, F3, 21) (dual of [6562, 6449, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(381, 6562, F3, 15) (dual of [6562, 6481, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(311, 41, F3, 5) (dual of [41, 30, 6]-code), using
- (u, u+v)-construction [i] based on
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- (u, u+v)-construction [i] based on
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3124, 6603, F3, 21) (dual of [6603, 6479, 22]-code), using
(103, 103+21, 1673151)-Net in Base 3 — Upper bound on s
There is no (103, 124, 1673152)-net in base 3, because
- 1 times m-reduction [i] would yield (103, 123, 1673152)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 48519 414487 992370 302343 909504 621790 114993 103273 948719 700609 > 3123 [i]