Best Known (39, 44, s)-Nets in Base 3
(39, 44, 2391492)-Net over F3 — Constructive and digital
Digital (39, 44, 2391492)-net over F3, using
- net defined by OOA [i] based on linear OOA(344, 2391492, F3, 5, 5) (dual of [(2391492, 5), 11957416, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(344, 4782985, F3, 5) (dual of [4782985, 4782941, 6]-code), using
- construction X4 applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(343, 4782969, F3, 5) (dual of [4782969, 4782926, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(329, 4782969, F3, 4) (dual of [4782969, 4782940, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [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(4) ⊂ Ce(3) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(344, 4782985, F3, 5) (dual of [4782985, 4782941, 6]-code), using
(39, 44, 4782985)-Net over F3 — Digital
Digital (39, 44, 4782985)-net over F3, using
- net defined by OOA [i] based on linear OOA(344, 4782985, F3, 5, 5) (dual of [(4782985, 5), 23914881, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(344, 4782985, F3, 4, 5) (dual of [(4782985, 4), 19131896, 6]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(344, 4782985, F3, 5) (dual of [4782985, 4782941, 6]-code), using
- construction X4 applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(343, 4782969, F3, 5) (dual of [4782969, 4782926, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(329, 4782969, F3, 4) (dual of [4782969, 4782940, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [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(4) ⊂ Ce(3) [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(344, 4782985, F3, 5) (dual of [4782985, 4782941, 6]-code), using
- appending kth column [i] based on linear OOA(344, 4782985, F3, 4, 5) (dual of [(4782985, 4), 19131896, 6]-NRT-code), using
(39, 44, large)-Net in Base 3 — Upper bound on s
There is no (39, 44, large)-net in base 3, because
- 3 times m-reduction [i] would yield (39, 41, large)-net in base 3, but