Best Known (112, 130, s)-Nets in Base 3
(112, 130, 6568)-Net over F3 — Constructive and digital
Digital (112, 130, 6568)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 7)-net over F3, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 1 and N(F) ≥ 7, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- digital (102, 120, 6561)-net over F3, using
- net defined by OOA [i] based on linear OOA(3120, 6561, F3, 18, 18) (dual of [(6561, 18), 117978, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(3120, 59049, F3, 18) (dual of [59049, 58929, 19]-code), using
- 1 times truncation [i] based on linear OA(3121, 59050, F3, 19) (dual of [59050, 58929, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(3121, 59050, F3, 19) (dual of [59050, 58929, 20]-code), using
- OA 9-folding and stacking [i] based on linear OA(3120, 59049, F3, 18) (dual of [59049, 58929, 19]-code), using
- net defined by OOA [i] based on linear OOA(3120, 6561, F3, 18, 18) (dual of [(6561, 18), 117978, 19]-NRT-code), using
- digital (1, 10, 7)-net over F3, using
(112, 130, 29549)-Net over F3 — Digital
Digital (112, 130, 29549)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3130, 29549, F3, 2, 18) (dual of [(29549, 2), 58968, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3130, 59098, F3, 18) (dual of [59098, 58968, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3130, 59099, F3, 18) (dual of [59099, 58969, 19]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- linear OA(3121, 59050, F3, 19) (dual of [59050, 58929, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(381, 59050, F3, 13) (dual of [59050, 58969, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(39, 49, F3, 4) (dual of [49, 40, 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 C([0,9]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3130, 59099, F3, 18) (dual of [59099, 58969, 19]-code), using
- OOA 2-folding [i] based on linear OA(3130, 59098, F3, 18) (dual of [59098, 58968, 19]-code), using
(112, 130, large)-Net in Base 3 — Upper bound on s
There is no (112, 130, large)-net in base 3, because
- 16 times m-reduction [i] would yield (112, 114, large)-net in base 3, but