Best Known (219, 219+31, s)-Nets in Base 3
(219, 219+31, 35432)-Net over F3 — Constructive and digital
Digital (219, 250, 35432)-net over F3, using
- 31 times duplication [i] based on digital (218, 249, 35432)-net over F3, using
- net defined by OOA [i] based on linear OOA(3249, 35432, F3, 31, 31) (dual of [(35432, 31), 1098143, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3249, 531481, F3, 31) (dual of [531481, 531232, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3249, 531482, F3, 31) (dual of [531482, 531233, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(25) [i] based on
- linear OA(3241, 531441, F3, 31) (dual of [531441, 531200, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3205, 531441, F3, 26) (dual of [531441, 531236, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- construction X applied to Ce(30) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(3249, 531482, F3, 31) (dual of [531482, 531233, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3249, 531481, F3, 31) (dual of [531481, 531232, 32]-code), using
- net defined by OOA [i] based on linear OOA(3249, 35432, F3, 31, 31) (dual of [(35432, 31), 1098143, 32]-NRT-code), using
(219, 219+31, 132871)-Net over F3 — Digital
Digital (219, 250, 132871)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3250, 132871, F3, 4, 31) (dual of [(132871, 4), 531234, 32]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3250, 531484, F3, 31) (dual of [531484, 531234, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3250, 531486, F3, 31) (dual of [531486, 531236, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(25) [i] based on
- linear OA(3241, 531441, F3, 31) (dual of [531441, 531200, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3205, 531441, F3, 26) (dual of [531441, 531236, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(39, 45, F3, 4) (dual of [45, 36, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- construction X applied to Ce(30) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(3250, 531486, F3, 31) (dual of [531486, 531236, 32]-code), using
- OOA 4-folding [i] based on linear OA(3250, 531484, F3, 31) (dual of [531484, 531234, 32]-code), using
(219, 219+31, large)-Net in Base 3 — Upper bound on s
There is no (219, 250, large)-net in base 3, because
- 29 times m-reduction [i] would yield (219, 221, large)-net in base 3, but