Best Known (178, 189, s)-Nets in Base 3
(178, 189, 6714164)-Net over F3 — Constructive and digital
Digital (178, 189, 6714164)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (20, 25, 3284)-net over F3, using
- net defined by OOA [i] based on linear OOA(325, 3284, F3, 5, 5) (dual of [(3284, 5), 16395, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(325, 6569, F3, 5) (dual of [6569, 6544, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(325, 6561, F3, 5) (dual of [6561, 6536, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(317, 6561, F3, 4) (dual of [6561, 6544, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(30, 8, F3, 0) (dual of [8, 8, 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
- OOA 2-folding and stacking with additional row [i] based on linear OA(325, 6569, F3, 5) (dual of [6569, 6544, 6]-code), using
- net defined by OOA [i] based on linear OOA(325, 3284, F3, 5, 5) (dual of [(3284, 5), 16395, 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 (20, 25, 3284)-net over F3, using
(178, 189, large)-Net over F3 — Digital
Digital (178, 189, large)-net over F3, using
- t-expansion [i] based on digital (175, 189, large)-net over F3, using
- 2 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
- 2 times m-reduction [i] based on digital (175, 191, large)-net over F3, using
(178, 189, large)-Net in Base 3 — Upper bound on s
There is no (178, 189, large)-net in base 3, because
- 9 times m-reduction [i] would yield (178, 180, large)-net in base 3, but