Best Known (44, 44+17, s)-Nets in Base 3
(44, 44+17, 156)-Net over F3 — Constructive and digital
Digital (44, 61, 156)-net over F3, using
- 31 times duplication [i] based on digital (43, 60, 156)-net over F3, using
- trace code for nets [i] based on digital (3, 20, 52)-net over F27, using
- net from sequence [i] based on digital (3, 51)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 3 and N(F) ≥ 52, using
- net from sequence [i] based on digital (3, 51)-sequence over F27, using
- trace code for nets [i] based on digital (3, 20, 52)-net over F27, using
(44, 44+17, 248)-Net over F3 — Digital
Digital (44, 61, 248)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(361, 248, F3, 17) (dual of [248, 187, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(361, 263, F3, 17) (dual of [263, 202, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- linear OA(356, 243, F3, 17) (dual of [243, 187, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(341, 243, F3, 13) (dual of [243, 202, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(35, 20, F3, 3) (dual of [20, 15, 4]-code or 20-cap in PG(4,3)), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(361, 263, F3, 17) (dual of [263, 202, 18]-code), using
(44, 44+17, 7122)-Net in Base 3 — Upper bound on s
There is no (44, 61, 7123)-net in base 3, because
- 1 times m-reduction [i] would yield (44, 60, 7123)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 42430 456281 703943 848407 357681 > 360 [i]