Best Known (62, 79, s)-Nets in Base 3
(62, 79, 464)-Net over F3 — Constructive and digital
Digital (62, 79, 464)-net over F3, using
- 1 times m-reduction [i] based on digital (62, 80, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 20, 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, 20, 116)-net over F81, using
(62, 79, 1097)-Net over F3 — Digital
Digital (62, 79, 1097)-net over F3, using
- 31 times duplication [i] based on digital (61, 78, 1097)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(378, 1097, F3, 2, 17) (dual of [(1097, 2), 2116, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(378, 2194, F3, 17) (dual of [2194, 2116, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(378, 2187, F3, 17) (dual of [2187, 2109, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(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(16) ⊂ Ce(15) [i] based on
- OOA 2-folding [i] based on linear OA(378, 2194, F3, 17) (dual of [2194, 2116, 18]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(378, 1097, F3, 2, 17) (dual of [(1097, 2), 2116, 18]-NRT-code), using
(62, 79, 84441)-Net in Base 3 — Upper bound on s
There is no (62, 79, 84442)-net in base 3, because
- 1 times m-reduction [i] would yield (62, 78, 84442)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 16 424648 763675 100761 900398 871566 683089 > 378 [i]