Best Known (61, 61+15, s)-Nets in Base 3
(61, 61+15, 600)-Net over F3 — Constructive and digital
Digital (61, 76, 600)-net over F3, using
- trace code for nets [i] based on digital (4, 19, 150)-net over F81, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 4 and N(F) ≥ 150, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
(61, 61+15, 1591)-Net over F3 — Digital
Digital (61, 76, 1591)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(376, 1591, F3, 15) (dual of [1591, 1515, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(376, 2209, F3, 15) (dual of [2209, 2133, 16]-code), using
- construction XX applied to Ce(15) ⊂ Ce(12) ⊂ Ce(10) [i] based on
- linear OA(371, 2187, F3, 16) (dual of [2187, 2116, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(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(31, 18, F3, 1) (dual of [18, 17, 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, 4, F3, 1) (dual of [4, 3, 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 (see above)
- construction XX applied to Ce(15) ⊂ Ce(12) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(376, 2209, F3, 15) (dual of [2209, 2133, 16]-code), using
(61, 61+15, 218720)-Net in Base 3 — Upper bound on s
There is no (61, 76, 218721)-net in base 3, because
- 1 times m-reduction [i] would yield (61, 75, 218721)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 608279 465741 390927 913797 448748 261899 > 375 [i]