Best Known (107−13, 107, s)-Nets in Base 3
(107−13, 107, 265723)-Net over F3 — Constructive and digital
Digital (94, 107, 265723)-net over F3, using
- net defined by OOA [i] based on linear OOA(3107, 265723, F3, 13, 13) (dual of [(265723, 13), 3454292, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3107, 1594339, F3, 13) (dual of [1594339, 1594232, 14]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3106, 1594338, F3, 13) (dual of [1594338, 1594232, 14]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(3105, 1594323, F3, 13) (dual of [1594323, 1594218, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(392, 1594323, F3, 11) (dual of [1594323, 1594231, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(314, 15, F3, 14) (dual of [15, 1, 15]-code or 15-arc in PG(13,3)), using
- dual of repetition code with length 15 [i]
- linear OA(31, 15, F3, 1) (dual of [15, 14, 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 code embedding in larger space [i] based on linear OA(3106, 1594338, F3, 13) (dual of [1594338, 1594232, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3107, 1594339, F3, 13) (dual of [1594339, 1594232, 14]-code), using
(107−13, 107, 531446)-Net over F3 — Digital
Digital (94, 107, 531446)-net over F3, using
- 31 times duplication [i] based on digital (93, 106, 531446)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3106, 531446, F3, 3, 13) (dual of [(531446, 3), 1594232, 14]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3106, 1594338, F3, 13) (dual of [1594338, 1594232, 14]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(3105, 1594323, F3, 13) (dual of [1594323, 1594218, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(392, 1594323, F3, 11) (dual of [1594323, 1594231, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(314, 15, F3, 14) (dual of [15, 1, 15]-code or 15-arc in PG(13,3)), using
- dual of repetition code with length 15 [i]
- linear OA(31, 15, F3, 1) (dual of [15, 14, 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
- OOA 3-folding [i] based on linear OA(3106, 1594338, F3, 13) (dual of [1594338, 1594232, 14]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3106, 531446, F3, 3, 13) (dual of [(531446, 3), 1594232, 14]-NRT-code), using
(107−13, 107, large)-Net in Base 3 — Upper bound on s
There is no (94, 107, large)-net in base 3, because
- 11 times m-reduction [i] would yield (94, 96, large)-net in base 3, but