Best Known (49, 49+12, s)-Nets in Base 3
(49, 49+12, 464)-Net over F3 — Constructive and digital
Digital (49, 61, 464)-net over F3, using
- 31 times duplication [i] based on digital (48, 60, 464)-net over F3, using
- t-expansion [i] based on digital (47, 60, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 15, 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, 15, 116)-net over F81, using
- t-expansion [i] based on digital (47, 60, 464)-net over F3, using
(49, 49+12, 1642)-Net over F3 — Digital
Digital (49, 61, 1642)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(361, 1642, F3, 12) (dual of [1642, 1581, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(361, 2207, F3, 12) (dual of [2207, 2146, 13]-code), using
- construction XX applied to Ce(12) ⊂ Ce(9) ⊂ Ce(7) [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(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(336, 2187, F3, 8) (dual of [2187, 2151, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(31, 17, F3, 1) (dual of [17, 16, 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(31, 3, F3, 1) (dual of [3, 2, 2]-code), using
- Reed–Solomon code RS(2,3) [i]
- construction XX applied to Ce(12) ⊂ Ce(9) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(361, 2207, F3, 12) (dual of [2207, 2146, 13]-code), using
(49, 49+12, 106145)-Net in Base 3 — Upper bound on s
There is no (49, 61, 106146)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 127175 744090 994910 968457 578493 > 361 [i]