Best Known (215, 215+27, s)-Nets in Base 3
(215, 215+27, 122643)-Net over F3 — Constructive and digital
Digital (215, 242, 122643)-net over F3, using
- 31 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
(215, 215+27, 427874)-Net over F3 — Digital
Digital (215, 242, 427874)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3242, 427874, F3, 3, 27) (dual of [(427874, 3), 1283380, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3242, 531456, F3, 3, 27) (dual of [(531456, 3), 1594126, 28]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3241, 531456, F3, 3, 27) (dual of [(531456, 3), 1594127, 28]-NRT-code), using
- OOA 3-folding [i] 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
- OOA 3-folding [i] based on linear OA(3241, 1594368, F3, 27) (dual of [1594368, 1594127, 28]-code), using
- 31 times duplication [i] based on linear OOA(3241, 531456, F3, 3, 27) (dual of [(531456, 3), 1594127, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3242, 531456, F3, 3, 27) (dual of [(531456, 3), 1594126, 28]-NRT-code), using
(215, 215+27, large)-Net in Base 3 — Upper bound on s
There is no (215, 242, large)-net in base 3, because
- 25 times m-reduction [i] would yield (215, 217, large)-net in base 3, but