Best Known (193, 193+43, s)-Nets in Base 3
(193, 193+43, 1480)-Net over F3 — Constructive and digital
Digital (193, 236, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 59, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
(193, 193+43, 4342)-Net over F3 — Digital
Digital (193, 236, 4342)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3236, 4342, F3, 43) (dual of [4342, 4106, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(3236, 6603, F3, 43) (dual of [6603, 6367, 44]-code), using
- construction X applied to C([0,21]) ⊂ C([0,18]) [i] based on
- linear OA(3225, 6562, F3, 43) (dual of [6562, 6337, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,21], and minimum distance d ≥ |{−21,−20,…,21}|+1 = 44 (BCH-bound) [i]
- linear OA(3193, 6562, F3, 37) (dual of [6562, 6369, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (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([0,21]) ⊂ C([0,18]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3236, 6603, F3, 43) (dual of [6603, 6367, 44]-code), using
(193, 193+43, 947692)-Net in Base 3 — Upper bound on s
There is no (193, 236, 947693)-net in base 3, because
- 1 times m-reduction [i] would yield (193, 235, 947693)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 13289 370359 954748 431119 041061 385965 346620 147999 033745 437906 775974 134816 155558 413101 189996 537275 849453 166778 345131 > 3235 [i]