Best Known (165−18, 165, s)-Nets in Base 3
(165−18, 165, 177152)-Net over F3 — Constructive and digital
Digital (147, 165, 177152)-net over F3, using
- 1 times m-reduction [i] based on digital (147, 166, 177152)-net over F3, using
- net defined by OOA [i] based on linear OOA(3166, 177152, F3, 19, 19) (dual of [(177152, 19), 3365722, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3166, 1594369, F3, 19) (dual of [1594369, 1594203, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3166, 1594371, F3, 19) (dual of [1594371, 1594205, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(3157, 1594323, F3, 19) (dual of [1594323, 1594166, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3118, 1594323, F3, 14) (dual of [1594323, 1594205, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(18) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(3166, 1594371, F3, 19) (dual of [1594371, 1594205, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3166, 1594369, F3, 19) (dual of [1594369, 1594203, 20]-code), using
- net defined by OOA [i] based on linear OOA(3166, 177152, F3, 19, 19) (dual of [(177152, 19), 3365722, 20]-NRT-code), using
(165−18, 165, 531456)-Net over F3 — Digital
Digital (147, 165, 531456)-net over F3, using
- 32 times duplication [i] based on digital (145, 163, 531456)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3163, 531456, F3, 3, 18) (dual of [(531456, 3), 1594205, 19]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3163, 1594368, F3, 18) (dual of [1594368, 1594205, 19]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(3157, 1594323, F3, 19) (dual of [1594323, 1594166, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3118, 1594323, F3, 14) (dual of [1594323, 1594205, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(18) ⊂ Ce(13) [i] based on
- OOA 3-folding [i] based on linear OA(3163, 1594368, F3, 18) (dual of [1594368, 1594205, 19]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3163, 531456, F3, 3, 18) (dual of [(531456, 3), 1594205, 19]-NRT-code), using
(165−18, 165, large)-Net in Base 3 — Upper bound on s
There is no (147, 165, large)-net in base 3, because
- 16 times m-reduction [i] would yield (147, 149, large)-net in base 3, but