Best Known (79, 79+20, s)-Nets in Base 3
(79, 79+20, 600)-Net over F3 — Constructive and digital
Digital (79, 99, 600)-net over F3, using
- 1 times m-reduction [i] based on digital (79, 100, 600)-net over F3, using
- trace code for nets [i] based on digital (4, 25, 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
- trace code for nets [i] based on digital (4, 25, 150)-net over F81, using
(79, 79+20, 1479)-Net over F3 — Digital
Digital (79, 99, 1479)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(399, 1479, F3, 20) (dual of [1479, 1380, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(399, 2215, F3, 20) (dual of [2215, 2116, 21]-code), using
- 1 times code embedding in larger space [i] based on linear OA(398, 2214, F3, 20) (dual of [2214, 2116, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(15) [i] based on
- linear OA(392, 2187, F3, 20) (dual of [2187, 2095, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- 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(36, 27, F3, 3) (dual of [27, 21, 4]-code or 27-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(19) ⊂ Ce(15) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(398, 2214, F3, 20) (dual of [2214, 2116, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(399, 2215, F3, 20) (dual of [2215, 2116, 21]-code), using
(79, 79+20, 119787)-Net in Base 3 — Upper bound on s
There is no (79, 99, 119788)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 171796 559769 322781 234570 060953 425208 522557 433257 > 399 [i]