Best Known (137, 137+16, s)-Nets in Base 3
(137, 137+16, 1048575)-Net over F3 — Constructive and digital
Digital (137, 153, 1048575)-net over F3, using
- 32 times duplication [i] based on digital (135, 151, 1048575)-net over F3, using
- net defined by OOA [i] based on linear OOA(3151, 1048575, F3, 16, 16) (dual of [(1048575, 16), 16777049, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(3151, 8388600, F3, 16) (dual of [8388600, 8388449, 17]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(3151, large, F3, 16) (dual of [large, large−151, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(3151, 8388600, F3, 16) (dual of [8388600, 8388449, 17]-code), using
- net defined by OOA [i] based on linear OOA(3151, 1048575, F3, 16, 16) (dual of [(1048575, 16), 16777049, 17]-NRT-code), using
(137, 137+16, 2434138)-Net over F3 — Digital
Digital (137, 153, 2434138)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3153, 2434138, F3, 3, 16) (dual of [(2434138, 3), 7302261, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3153, 2796201, F3, 3, 16) (dual of [(2796201, 3), 8388450, 17]-NRT-code), using
- 32 times duplication [i] based on linear OOA(3151, 2796201, F3, 3, 16) (dual of [(2796201, 3), 8388452, 17]-NRT-code), using
- OOA 3-folding [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]
- OOA 3-folding [i] based on linear OA(3151, large, F3, 16) (dual of [large, large−151, 17]-code), using
- 32 times duplication [i] based on linear OOA(3151, 2796201, F3, 3, 16) (dual of [(2796201, 3), 8388452, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3153, 2796201, F3, 3, 16) (dual of [(2796201, 3), 8388450, 17]-NRT-code), using
(137, 137+16, large)-Net in Base 3 — Upper bound on s
There is no (137, 153, large)-net in base 3, because
- 14 times m-reduction [i] would yield (137, 139, large)-net in base 3, but