Best Known (56−11, 56, s)-Nets in Base 3
(56−11, 56, 464)-Net over F3 — Constructive and digital
Digital (45, 56, 464)-net over F3, using
- t-expansion [i] based on digital (44, 56, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 14, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- trace code for nets [i] based on digital (2, 14, 116)-net over F81, using
(56−11, 56, 1700)-Net over F3 — Digital
Digital (45, 56, 1700)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(356, 1700, F3, 11) (dual of [1700, 1644, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(356, 2199, F3, 11) (dual of [2199, 2143, 12]-code), using
- (u, u+v)-construction [i] based on
- linear OA(36, 12, F3, 5) (dual of [12, 6, 6]-code), using
- extended Golay code Ge(3) [i]
- linear OA(350, 2187, F3, 11) (dual of [2187, 2137, 12]-code), using
- an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(36, 12, F3, 5) (dual of [12, 6, 6]-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(356, 2199, F3, 11) (dual of [2199, 2143, 12]-code), using
(56−11, 56, 230744)-Net in Base 3 — Upper bound on s
There is no (45, 56, 230745)-net in base 3, because
- 1 times m-reduction [i] would yield (45, 55, 230745)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 174 450643 251924 111486 909219 > 355 [i]