Best Known (52, 52+7, s)-Nets in Base 3
(52, 52+7, 1594328)-Net over F3 — Constructive and digital
Digital (52, 59, 1594328)-net over F3, using
- 31 times duplication [i] based on digital (51, 58, 1594328)-net over F3, using
- net defined by OOA [i] based on linear OOA(358, 1594328, F3, 7, 7) (dual of [(1594328, 7), 11160238, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(358, 4782985, F3, 7) (dual of [4782985, 4782927, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(357, 4782969, F3, 7) (dual of [4782969, 4782912, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- 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(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(6) ⊂ Ce(4) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(358, 4782985, F3, 7) (dual of [4782985, 4782927, 8]-code), using
- net defined by OOA [i] based on linear OOA(358, 1594328, F3, 7, 7) (dual of [(1594328, 7), 11160238, 8]-NRT-code), using
(52, 52+7, 2391493)-Net over F3 — Digital
Digital (52, 59, 2391493)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(359, 2391493, F3, 2, 7) (dual of [(2391493, 2), 4782927, 8]-NRT-code), using
- OOA 2-folding [i] based on linear OA(359, 4782986, F3, 7) (dual of [4782986, 4782927, 8]-code), using
- 1 times code embedding in larger space [i] based on linear OA(358, 4782985, F3, 7) (dual of [4782985, 4782927, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(357, 4782969, F3, 7) (dual of [4782969, 4782912, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- 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(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(6) ⊂ Ce(4) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(358, 4782985, F3, 7) (dual of [4782985, 4782927, 8]-code), using
- OOA 2-folding [i] based on linear OA(359, 4782986, F3, 7) (dual of [4782986, 4782927, 8]-code), using
(52, 52+7, large)-Net in Base 3 — Upper bound on s
There is no (52, 59, large)-net in base 3, because
- 5 times m-reduction [i] would yield (52, 54, large)-net in base 3, but