Best Known (88−13, 88, s)-Nets in Base 3
(88−13, 88, 9848)-Net over F3 — Constructive and digital
Digital (75, 88, 9848)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 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
- a shift-net [i]
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- digital (68, 81, 9841)-net over F3, using
- net defined by OOA [i] based on linear OOA(381, 9841, F3, 13, 13) (dual of [(9841, 13), 127852, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(381, 59047, F3, 13) (dual of [59047, 58966, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(381, 59049, F3, 13) (dual of [59049, 58968, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(381, 59049, F3, 13) (dual of [59049, 58968, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(381, 59047, F3, 13) (dual of [59047, 58966, 14]-code), using
- net defined by OOA [i] based on linear OOA(381, 9841, F3, 13, 13) (dual of [(9841, 13), 127852, 14]-NRT-code), using
- digital (1, 7, 7)-net over F3, using
(88−13, 88, 28711)-Net over F3 — Digital
Digital (75, 88, 28711)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(388, 28711, F3, 2, 13) (dual of [(28711, 2), 57334, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(388, 29538, F3, 2, 13) (dual of [(29538, 2), 58988, 14]-NRT-code), using
- 31 times duplication [i] based on linear OOA(387, 29538, F3, 2, 13) (dual of [(29538, 2), 58989, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(387, 59076, F3, 13) (dual of [59076, 58989, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- 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(361, 59050, F3, 9) (dual of [59050, 58989, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(36, 26, F3, 3) (dual of [26, 20, 4]-code or 26-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 C([0,6]) ⊂ C([0,4]) [i] based on
- OOA 2-folding [i] based on linear OA(387, 59076, F3, 13) (dual of [59076, 58989, 14]-code), using
- 31 times duplication [i] based on linear OOA(387, 29538, F3, 2, 13) (dual of [(29538, 2), 58989, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(388, 29538, F3, 2, 13) (dual of [(29538, 2), 58988, 14]-NRT-code), using
(88−13, 88, large)-Net in Base 3 — Upper bound on s
There is no (75, 88, large)-net in base 3, because
- 11 times m-reduction [i] would yield (75, 77, large)-net in base 3, but