Best Known (43−11, 43, s)-Nets in Base 3
(43−11, 43, 164)-Net over F3 — Constructive and digital
Digital (32, 43, 164)-net over F3, using
- base reduction for projective spaces (embedding PG(21,9) in PG(42,3)) for nets [i] based on digital (11, 22, 164)-net over F9, using
- trace code for nets [i] based on digital (0, 11, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- trace code for nets [i] based on digital (0, 11, 82)-net over F81, using
(43−11, 43, 367)-Net over F3 — Digital
Digital (32, 43, 367)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(343, 367, F3, 2, 11) (dual of [(367, 2), 691, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(343, 734, F3, 11) (dual of [734, 691, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(343, 735, F3, 11) (dual of [735, 692, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(343, 729, F3, 11) (dual of [729, 686, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(337, 729, F3, 10) (dual of [729, 692, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(30, 6, F3, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(343, 735, F3, 11) (dual of [735, 692, 12]-code), using
- OOA 2-folding [i] based on linear OA(343, 734, F3, 11) (dual of [734, 691, 12]-code), using
(43−11, 43, 13257)-Net in Base 3 — Upper bound on s
There is no (32, 43, 13258)-net in base 3, because
- 1 times m-reduction [i] would yield (32, 42, 13258)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 109 419723 223507 581165 > 342 [i]