Best Known (113−12, 113, s)-Nets in Base 3
(113−12, 113, 797164)-Net over F3 — Constructive and digital
Digital (101, 113, 797164)-net over F3, using
- net defined by OOA [i] based on linear OOA(3113, 797164, F3, 12, 12) (dual of [(797164, 12), 9565855, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(3113, 4782984, F3, 12) (dual of [4782984, 4782871, 13]-code), using
- 1 times truncation [i] based on linear OA(3114, 4782985, F3, 13) (dual of [4782985, 4782871, 14]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(3113, 4782969, F3, 13) (dual of [4782969, 4782856, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(399, 4782969, F3, 11) (dual of [4782969, 4782870, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(315, 16, F3, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,3)), using
- dual of repetition code with length 16 [i]
- linear OA(31, 16, F3, 1) (dual of [16, 15, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- 1 times truncation [i] based on linear OA(3114, 4782985, F3, 13) (dual of [4782985, 4782871, 14]-code), using
- OA 6-folding and stacking [i] based on linear OA(3113, 4782984, F3, 12) (dual of [4782984, 4782871, 13]-code), using
(113−12, 113, 1594328)-Net over F3 — Digital
Digital (101, 113, 1594328)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3113, 1594328, F3, 3, 12) (dual of [(1594328, 3), 4782871, 13]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3113, 4782984, F3, 12) (dual of [4782984, 4782871, 13]-code), using
- 1 times truncation [i] based on linear OA(3114, 4782985, F3, 13) (dual of [4782985, 4782871, 14]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(3113, 4782969, F3, 13) (dual of [4782969, 4782856, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(399, 4782969, F3, 11) (dual of [4782969, 4782870, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(315, 16, F3, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,3)), using
- dual of repetition code with length 16 [i]
- linear OA(31, 16, F3, 1) (dual of [16, 15, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- 1 times truncation [i] based on linear OA(3114, 4782985, F3, 13) (dual of [4782985, 4782871, 14]-code), using
- OOA 3-folding [i] based on linear OA(3113, 4782984, F3, 12) (dual of [4782984, 4782871, 13]-code), using
(113−12, 113, large)-Net in Base 3 — Upper bound on s
There is no (101, 113, large)-net in base 3, because
- 10 times m-reduction [i] would yield (101, 103, large)-net in base 3, but