Best Known (112−9, 112, s)-Nets in Base 3
(112−9, 112, 2391491)-Net over F3 — Constructive and digital
Digital (103, 112, 2391491)-net over F3, using
- net defined by OOA [i] based on linear OOA(3112, 2391491, F3, 12, 9) (dual of [(2391491, 12), 28697780, 10]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(3112, 2391492, F3, 4, 9) (dual of [(2391492, 4), 9565856, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(327, 1594322, F3, 4, 4) (dual of [(1594322, 4), 6377261, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(327, 1594322, F3, 3, 4) (dual of [(1594322, 3), 4782939, 5]-NRT-code), using
- extracting embedded OOA [i] based on digital (23, 27, 1594322)-net over F3, using
- appending kth column [i] based on linear OOA(327, 1594322, F3, 3, 4) (dual of [(1594322, 3), 4782939, 5]-NRT-code), using
- linear OOA(385, 1195746, F3, 4, 9) (dual of [(1195746, 4), 4782899, 10]-NRT-code), using
- OOA 4-folding [i] based on linear OA(385, 4782984, F3, 9) (dual of [4782984, 4782899, 10]-code), using
- 1 times truncation [i] based on linear OA(386, 4782985, F3, 10) (dual of [4782985, 4782899, 11]-code), using
- construction X4 applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(385, 4782969, F3, 10) (dual of [4782969, 4782884, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(371, 4782969, F3, 8) (dual of [4782969, 4782898, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(315, 16, F3, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,3)), using
- dual of repetition code with length 16 [i]
- linear OA(31, 16, F3, 1) (dual of [16, 15, 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 X4 applied to Ce(9) ⊂ Ce(7) [i] based on
- 1 times truncation [i] based on linear OA(386, 4782985, F3, 10) (dual of [4782985, 4782899, 11]-code), using
- OOA 4-folding [i] based on linear OA(385, 4782984, F3, 9) (dual of [4782984, 4782899, 10]-code), using
- linear OOA(327, 1594322, F3, 4, 4) (dual of [(1594322, 4), 6377261, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(3112, 2391492, F3, 4, 9) (dual of [(2391492, 4), 9565856, 10]-NRT-code), using
(112−9, 112, large)-Net over F3 — Digital
Digital (103, 112, large)-net over F3, using
- 1 times m-reduction [i] based on digital (103, 113, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3113, large, F3, 10) (dual of [large, large−113, 11]-code), using
- 22 times code embedding in larger space [i] based on linear OA(391, large, F3, 10) (dual of [large, large−91, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- 22 times code embedding in larger space [i] based on linear OA(391, large, F3, 10) (dual of [large, large−91, 11]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3113, large, F3, 10) (dual of [large, large−113, 11]-code), using
(112−9, 112, large)-Net in Base 3 — Upper bound on s
There is no (103, 112, large)-net in base 3, because
- 7 times m-reduction [i] would yield (103, 105, large)-net in base 3, but