Best Known (72−16, 72, s)-Nets in Base 3
(72−16, 72, 464)-Net over F3 — Constructive and digital
Digital (56, 72, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 18, 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
(72−16, 72, 1039)-Net over F3 — Digital
Digital (56, 72, 1039)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(372, 1039, F3, 2, 16) (dual of [(1039, 2), 2006, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(372, 1097, F3, 2, 16) (dual of [(1097, 2), 2122, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(372, 2194, F3, 16) (dual of [2194, 2122, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(372, 2195, F3, 16) (dual of [2195, 2123, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [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(364, 2187, F3, 14) (dual of [2187, 2123, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(31, 8, F3, 1) (dual of [8, 7, 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
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(372, 2195, F3, 16) (dual of [2195, 2123, 17]-code), using
- OOA 2-folding [i] based on linear OA(372, 2194, F3, 16) (dual of [2194, 2122, 17]-code), using
- discarding factors / shortening the dual code based on linear OOA(372, 1097, F3, 2, 16) (dual of [(1097, 2), 2122, 17]-NRT-code), using
(72−16, 72, 37039)-Net in Base 3 — Upper bound on s
There is no (56, 72, 37040)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 22531 527525 205029 446965 309057 980417 > 372 [i]