Best Known (64−14, 64, s)-Nets in Base 3
(64−14, 64, 464)-Net over F3 — Constructive and digital
Digital (50, 64, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 16, 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
(64−14, 64, 1097)-Net over F3 — Digital
Digital (50, 64, 1097)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(364, 1097, F3, 2, 14) (dual of [(1097, 2), 2130, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(364, 2194, F3, 14) (dual of [2194, 2130, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(364, 2187, F3, 14) (dual of [2187, 2123, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(357, 2187, F3, 13) (dual of [2187, 2130, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(30, 7, F3, 0) (dual of [7, 7, 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(13) ⊂ Ce(12) [i] based on
- OOA 2-folding [i] based on linear OA(364, 2194, F3, 14) (dual of [2194, 2130, 15]-code), using
(64−14, 64, 38910)-Net in Base 3 — Upper bound on s
There is no (50, 64, 38911)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 3 433949 753727 359624 179146 612723 > 364 [i]