Best Known (59−13, 59, s)-Nets in Base 3
(59−13, 59, 400)-Net over F3 — Constructive and digital
Digital (46, 59, 400)-net over F3, using
- 1 times m-reduction [i] based on digital (46, 60, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 15, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- trace code for nets [i] based on digital (1, 15, 100)-net over F81, using
(59−13, 59, 1098)-Net over F3 — Digital
Digital (46, 59, 1098)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(359, 1098, F3, 2, 13) (dual of [(1098, 2), 2137, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(359, 2196, F3, 13) (dual of [2196, 2137, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(359, 2197, F3, 13) (dual of [2197, 2138, 14]-code), using
- construction XX applied to Ce(12) ⊂ Ce(10) ⊂ Ce(9) [i] based on
- 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(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(343, 2187, F3, 10) (dual of [2187, 2144, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(31, 9, F3, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(12) ⊂ Ce(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(359, 2197, F3, 13) (dual of [2197, 2138, 14]-code), using
- OOA 2-folding [i] based on linear OA(359, 2196, F3, 13) (dual of [2196, 2137, 14]-code), using
(59−13, 59, 61280)-Net in Base 3 — Upper bound on s
There is no (46, 59, 61281)-net in base 3, because
- 1 times m-reduction [i] would yield (46, 58, 61281)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 4710 167904 578180 088986 739865 > 358 [i]