Best Known (137, 137+20, s)-Nets in Base 3
(137, 137+20, 53145)-Net over F3 — Constructive and digital
Digital (137, 157, 53145)-net over F3, using
- net defined by OOA [i] based on linear OOA(3157, 53145, F3, 20, 20) (dual of [(53145, 20), 1062743, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(3157, 531450, F3, 20) (dual of [531450, 531293, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3157, 531453, F3, 20) (dual of [531453, 531296, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(3157, 531441, F3, 20) (dual of [531441, 531284, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3145, 531441, F3, 19) (dual of [531441, 531296, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(30, 12, F3, 0) (dual of [12, 12, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(3157, 531453, F3, 20) (dual of [531453, 531296, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(3157, 531450, F3, 20) (dual of [531450, 531293, 21]-code), using
(137, 137+20, 132969)-Net over F3 — Digital
Digital (137, 157, 132969)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3157, 132969, F3, 3, 20) (dual of [(132969, 3), 398750, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3157, 177151, F3, 3, 20) (dual of [(177151, 3), 531296, 21]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3157, 531453, F3, 20) (dual of [531453, 531296, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(3157, 531441, F3, 20) (dual of [531441, 531284, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3145, 531441, F3, 19) (dual of [531441, 531296, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(30, 12, F3, 0) (dual of [12, 12, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- OOA 3-folding [i] based on linear OA(3157, 531453, F3, 20) (dual of [531453, 531296, 21]-code), using
- discarding factors / shortening the dual code based on linear OOA(3157, 177151, F3, 3, 20) (dual of [(177151, 3), 531296, 21]-NRT-code), using
(137, 137+20, large)-Net in Base 3 — Upper bound on s
There is no (137, 157, large)-net in base 3, because
- 18 times m-reduction [i] would yield (137, 139, large)-net in base 3, but