Best Known (44, 55, s)-Nets in Base 3
(44, 55, 464)-Net over F3 — Constructive and digital
Digital (44, 55, 464)-net over F3, using
- 1 times m-reduction [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
(44, 55, 1504)-Net over F3 — Digital
Digital (44, 55, 1504)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(355, 1504, F3, 11) (dual of [1504, 1449, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(355, 2193, F3, 11) (dual of [2193, 2138, 12]-code), using
- (u, u+v)-construction [i] based on
- linear OA(35, 6, F3, 5) (dual of [6, 1, 6]-code or 6-arc in PG(4,3)), using
- dual of repetition code with length 6 [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(35, 6, F3, 5) (dual of [6, 1, 6]-code or 6-arc in PG(4,3)), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(355, 2193, F3, 11) (dual of [2193, 2138, 12]-code), using
(44, 55, 185227)-Net in Base 3 — Upper bound on s
There is no (44, 55, 185228)-net in base 3, because
- 1 times m-reduction [i] would yield (44, 54, 185228)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 58 150663 306566 021439 613433 > 354 [i]