Best Known (159, 159+19, s)-Nets in Base 3
(159, 159+19, 531446)-Net over F3 — Constructive and digital
Digital (159, 178, 531446)-net over F3, using
- net defined by OOA [i] based on linear OOA(3178, 531446, F3, 19, 19) (dual of [(531446, 19), 10097296, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3178, 4783015, F3, 19) (dual of [4783015, 4782837, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3178, 4783020, F3, 19) (dual of [4783020, 4782842, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- 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(3127, 4782969, F3, 14) (dual of [4782969, 4782842, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(39, 51, F3, 4) (dual of [51, 42, 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(18) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(3178, 4783020, F3, 19) (dual of [4783020, 4782842, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3178, 4783015, F3, 19) (dual of [4783015, 4782837, 20]-code), using
(159, 159+19, 1195755)-Net over F3 — Digital
Digital (159, 178, 1195755)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3178, 1195755, F3, 4, 19) (dual of [(1195755, 4), 4782842, 20]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3178, 4783020, F3, 19) (dual of [4783020, 4782842, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- 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(3127, 4782969, F3, 14) (dual of [4782969, 4782842, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(39, 51, F3, 4) (dual of [51, 42, 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(18) ⊂ Ce(13) [i] based on
- OOA 4-folding [i] based on linear OA(3178, 4783020, F3, 19) (dual of [4783020, 4782842, 20]-code), using
(159, 159+19, large)-Net in Base 3 — Upper bound on s
There is no (159, 178, large)-net in base 3, because
- 17 times m-reduction [i] would yield (159, 161, large)-net in base 3, but