Best Known (172, 172+20, s)-Nets in Base 3
(172, 172+20, 478302)-Net over F3 — Constructive and digital
Digital (172, 192, 478302)-net over F3, using
- net defined by OOA [i] based on linear OOA(3192, 478302, F3, 20, 20) (dual of [(478302, 20), 9565848, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(3192, 4783020, F3, 20) (dual of [4783020, 4782828, 21]-code), using
- 3 times code embedding in larger space [i] based on linear OA(3189, 4783017, F3, 20) (dual of [4783017, 4782828, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(15) [i] based on
- linear OA(3183, 4782969, F3, 20) (dual of [4782969, 4782786, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3141, 4782969, F3, 16) (dual of [4782969, 4782828, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(19) ⊂ Ce(15) [i] based on
- 3 times code embedding in larger space [i] based on linear OA(3189, 4783017, F3, 20) (dual of [4783017, 4782828, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(3192, 4783020, F3, 20) (dual of [4783020, 4782828, 21]-code), using
(172, 172+20, 1470615)-Net over F3 — Digital
Digital (172, 192, 1470615)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3192, 1470615, F3, 3, 20) (dual of [(1470615, 3), 4411653, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3192, 1594340, F3, 3, 20) (dual of [(1594340, 3), 4782828, 21]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3189, 1594339, F3, 3, 20) (dual of [(1594339, 3), 4782828, 21]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3189, 4783017, F3, 20) (dual of [4783017, 4782828, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(15) [i] based on
- linear OA(3183, 4782969, F3, 20) (dual of [4782969, 4782786, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3141, 4782969, F3, 16) (dual of [4782969, 4782828, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(19) ⊂ Ce(15) [i] based on
- OOA 3-folding [i] based on linear OA(3189, 4783017, F3, 20) (dual of [4783017, 4782828, 21]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3189, 1594339, F3, 3, 20) (dual of [(1594339, 3), 4782828, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3192, 1594340, F3, 3, 20) (dual of [(1594340, 3), 4782828, 21]-NRT-code), using
(172, 172+20, large)-Net in Base 3 — Upper bound on s
There is no (172, 192, large)-net in base 3, because
- 18 times m-reduction [i] would yield (172, 174, large)-net in base 3, but