Best Known (219−23, 219, s)-Nets in Base 3
(219−23, 219, 434819)-Net over F3 — Constructive and digital
Digital (196, 219, 434819)-net over F3, using
- 32 times duplication [i] based on digital (194, 217, 434819)-net over F3, using
- net defined by OOA [i] based on linear OOA(3217, 434819, F3, 23, 23) (dual of [(434819, 23), 10000620, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3217, 4783010, F3, 23) (dual of [4783010, 4782793, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3217, 4783017, F3, 23) (dual of [4783017, 4782800, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(3211, 4782969, F3, 23) (dual of [4782969, 4782758, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3169, 4782969, F3, 19) (dual of [4782969, 4782800, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 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(22) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(3217, 4783017, F3, 23) (dual of [4783017, 4782800, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3217, 4783010, F3, 23) (dual of [4783010, 4782793, 24]-code), using
- net defined by OOA [i] based on linear OOA(3217, 434819, F3, 23, 23) (dual of [(434819, 23), 10000620, 24]-NRT-code), using
(219−23, 219, 1195754)-Net over F3 — Digital
Digital (196, 219, 1195754)-net over F3, using
- 32 times duplication [i] based on digital (194, 217, 1195754)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3217, 1195754, F3, 4, 23) (dual of [(1195754, 4), 4782799, 24]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3217, 4783016, F3, 23) (dual of [4783016, 4782799, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3217, 4783017, F3, 23) (dual of [4783017, 4782800, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(3211, 4782969, F3, 23) (dual of [4782969, 4782758, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3169, 4782969, F3, 19) (dual of [4782969, 4782800, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 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(22) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(3217, 4783017, F3, 23) (dual of [4783017, 4782800, 24]-code), using
- OOA 4-folding [i] based on linear OA(3217, 4783016, F3, 23) (dual of [4783016, 4782799, 24]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3217, 1195754, F3, 4, 23) (dual of [(1195754, 4), 4782799, 24]-NRT-code), using
(219−23, 219, large)-Net in Base 3 — Upper bound on s
There is no (196, 219, large)-net in base 3, because
- 21 times m-reduction [i] would yield (196, 198, large)-net in base 3, but