Best Known (179, 179+14, s)-Nets in Base 3
(179, 179+14, 2396742)-Net over F3 — Constructive and digital
Digital (179, 193, 2396742)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (50, 57, 1594323)-net over F3, using
- net defined by OOA [i] based on linear OOA(357, 1594323, F3, 7, 7) (dual of [(1594323, 7), 11160204, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(357, 4782970, F3, 7) (dual of [4782970, 4782913, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4782970 | 328−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- OOA 3-folding and stacking with additional row [i] based on linear OA(357, 4782970, F3, 7) (dual of [4782970, 4782913, 8]-code), using
- net defined by OOA [i] based on linear OOA(357, 1594323, F3, 7, 7) (dual of [(1594323, 7), 11160204, 8]-NRT-code), using
- digital (122, 136, 1198371)-net over F3, using
- net defined by OOA [i] based on linear OOA(3136, 1198371, F3, 14, 14) (dual of [(1198371, 14), 16777058, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(3136, 8388597, F3, 14) (dual of [8388597, 8388461, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(3136, large, F3, 14) (dual of [large, large−136, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(3136, large, F3, 14) (dual of [large, large−136, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(3136, 8388597, F3, 14) (dual of [8388597, 8388461, 15]-code), using
- net defined by OOA [i] based on linear OOA(3136, 1198371, F3, 14, 14) (dual of [(1198371, 14), 16777058, 15]-NRT-code), using
- digital (50, 57, 1594323)-net over F3, using
(179, 179+14, large)-Net over F3 — Digital
Digital (179, 193, large)-net over F3, using
- 32 times duplication [i] based on digital (177, 191, large)-net over F3, using
- t-expansion [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
- t-expansion [i] based on digital (175, 191, large)-net over F3, using
(179, 179+14, large)-Net in Base 3 — Upper bound on s
There is no (179, 193, large)-net in base 3, because
- 12 times m-reduction [i] would yield (179, 181, large)-net in base 3, but