Best Known (131−19, 131, s)-Nets in Base 3
(131−19, 131, 6568)-Net over F3 — Constructive and digital
Digital (112, 131, 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, 121, 6561)-net over F3, using
- net defined by OOA [i] based on linear OOA(3121, 6561, F3, 19, 19) (dual of [(6561, 19), 124538, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [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]
- OOA 9-folding and stacking with additional row [i] based on linear OA(3121, 59050, F3, 19) (dual of [59050, 58929, 20]-code), using
- net defined by OOA [i] based on linear OOA(3121, 6561, F3, 19, 19) (dual of [(6561, 19), 124538, 20]-NRT-code), using
- digital (1, 10, 7)-net over F3, using
(131−19, 131, 23880)-Net over F3 — Digital
Digital (112, 131, 23880)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3131, 23880, F3, 2, 19) (dual of [(23880, 2), 47629, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3131, 29544, F3, 2, 19) (dual of [(29544, 2), 58957, 20]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3130, 29544, F3, 2, 19) (dual of [(29544, 2), 58958, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3130, 59088, F3, 19) (dual of [59088, 58958, 20]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3129, 59087, F3, 19) (dual of [59087, 58958, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(3121, 59049, F3, 19) (dual of [59049, 58928, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(391, 59049, F3, 14) (dual of [59049, 58958, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(38, 38, F3, 4) (dual of [38, 30, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3129, 59087, F3, 19) (dual of [59087, 58958, 20]-code), using
- OOA 2-folding [i] based on linear OA(3130, 59088, F3, 19) (dual of [59088, 58958, 20]-code), using
- 31 times duplication [i] based on linear OOA(3130, 29544, F3, 2, 19) (dual of [(29544, 2), 58958, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3131, 29544, F3, 2, 19) (dual of [(29544, 2), 58957, 20]-NRT-code), using
(131−19, 131, large)-Net in Base 3 — Upper bound on s
There is no (112, 131, large)-net in base 3, because
- 17 times m-reduction [i] would yield (112, 114, large)-net in base 3, but