Best Known (186−11, 186, s)-Nets in Base 3
(186−11, 186, 6711976)-Net over F3 — Constructive and digital
Digital (175, 186, 6711976)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (17, 22, 1096)-net over F3, using
- net defined by OOA [i] based on linear OOA(322, 1096, F3, 5, 5) (dual of [(1096, 5), 5458, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(322, 2193, F3, 5) (dual of [2193, 2171, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(322, 2194, F3, 5) (dual of [2194, 2172, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(322, 2187, F3, 5) (dual of [2187, 2165, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(315, 2187, F3, 4) (dual of [2187, 2172, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [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(4) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(322, 2194, F3, 5) (dual of [2194, 2172, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(322, 2193, F3, 5) (dual of [2193, 2171, 6]-code), using
- net defined by OOA [i] based on linear OOA(322, 1096, F3, 5, 5) (dual of [(1096, 5), 5458, 6]-NRT-code), using
- digital (153, 164, 6710880)-net over F3, using
- trace code for nets [i] based on digital (30, 41, 1677720)-net over F81, using
- net defined by OOA [i] based on linear OOA(8141, 1677720, F81, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(8141, 8388601, F81, 11) (dual of [8388601, 8388560, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(8141, large, F81, 11) (dual of [large, large−41, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8141, large, F81, 11) (dual of [large, large−41, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(8141, 8388601, F81, 11) (dual of [8388601, 8388560, 12]-code), using
- net defined by OOA [i] based on linear OOA(8141, 1677720, F81, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- trace code for nets [i] based on digital (30, 41, 1677720)-net over F81, using
- digital (17, 22, 1096)-net over F3, using
(186−11, 186, large)-Net over F3 — Digital
Digital (175, 186, large)-net over F3, using
- 5 times m-reduction [i] based on digital (175, 191, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3191, large, F3, 16) (dual of [large, large−191, 17]-code), using
- 40 times code embedding in larger space [i] based on linear OA(3151, large, F3, 16) (dual of [large, large−151, 17]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 40 times code embedding in larger space [i] based on linear OA(3151, large, F3, 16) (dual of [large, large−151, 17]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3191, large, F3, 16) (dual of [large, large−191, 17]-code), using
(186−11, 186, large)-Net in Base 3 — Upper bound on s
There is no (175, 186, large)-net in base 3, because
- 9 times m-reduction [i] would yield (175, 177, large)-net in base 3, but