Best Known (216, 216+27, s)-Nets in Base 3
(216, 216+27, 122643)-Net over F3 — Constructive and digital
Digital (216, 243, 122643)-net over F3, using
- 32 times duplication [i] based on digital (214, 241, 122643)-net over F3, using
- net defined by OOA [i] based on linear OOA(3241, 122643, F3, 27, 27) (dual of [(122643, 27), 3311120, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3241, 1594360, F3, 27) (dual of [1594360, 1594119, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3241, 1594368, F3, 27) (dual of [1594368, 1594127, 28]-code), using
- construction X applied to Ce(27) ⊂ Ce(22) [i] based on
- linear OA(3235, 1594323, F3, 28) (dual of [1594323, 1594088, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3196, 1594323, F3, 23) (dual of [1594323, 1594127, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(36, 45, F3, 3) (dual of [45, 39, 4]-code or 45-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on 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(27) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(3241, 1594368, F3, 27) (dual of [1594368, 1594127, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3241, 1594360, F3, 27) (dual of [1594360, 1594119, 28]-code), using
- net defined by OOA [i] based on linear OOA(3241, 122643, F3, 27, 27) (dual of [(122643, 27), 3311120, 28]-NRT-code), using
(216, 216+27, 448809)-Net over F3 — Digital
Digital (216, 243, 448809)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3243, 448809, F3, 3, 27) (dual of [(448809, 3), 1346184, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3243, 531456, F3, 3, 27) (dual of [(531456, 3), 1594125, 28]-NRT-code), using
- 1 step truncation [i] based on linear OOA(3244, 531457, F3, 3, 28) (dual of [(531457, 3), 1594127, 29]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3244, 1594371, F3, 28) (dual of [1594371, 1594127, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(22) [i] based on
- linear OA(3235, 1594323, F3, 28) (dual of [1594323, 1594088, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3196, 1594323, F3, 23) (dual of [1594323, 1594127, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(39, 48, F3, 4) (dual of [48, 39, 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(27) ⊂ Ce(22) [i] based on
- OOA 3-folding [i] based on linear OA(3244, 1594371, F3, 28) (dual of [1594371, 1594127, 29]-code), using
- 1 step truncation [i] based on linear OOA(3244, 531457, F3, 3, 28) (dual of [(531457, 3), 1594127, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3243, 531456, F3, 3, 27) (dual of [(531456, 3), 1594125, 28]-NRT-code), using
(216, 216+27, large)-Net in Base 3 — Upper bound on s
There is no (216, 243, large)-net in base 3, because
- 25 times m-reduction [i] would yield (216, 218, large)-net in base 3, but