Best Known (174−24, 174, s)-Nets in Base 3
(174−24, 174, 4928)-Net over F3 — Constructive and digital
Digital (150, 174, 4928)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 14, 8)-net over F3, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 2 and N(F) ≥ 8, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- digital (136, 160, 4920)-net over F3, using
- net defined by OOA [i] based on linear OOA(3160, 4920, F3, 24, 24) (dual of [(4920, 24), 117920, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(3160, 59040, F3, 24) (dual of [59040, 58880, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3160, 59049, F3, 24) (dual of [59049, 58889, 25]-code), using
- 1 times truncation [i] based on linear OA(3161, 59050, F3, 25) (dual of [59050, 58889, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(3161, 59050, F3, 25) (dual of [59050, 58889, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3160, 59049, F3, 24) (dual of [59049, 58889, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(3160, 59040, F3, 24) (dual of [59040, 58880, 25]-code), using
- net defined by OOA [i] based on linear OOA(3160, 4920, F3, 24, 24) (dual of [(4920, 24), 117920, 25]-NRT-code), using
- digital (2, 14, 8)-net over F3, using
(174−24, 174, 29552)-Net over F3 — Digital
Digital (150, 174, 29552)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3174, 29552, F3, 2, 24) (dual of [(29552, 2), 58930, 25]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3174, 59104, F3, 24) (dual of [59104, 58930, 25]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(3170, 59099, F3, 24) (dual of [59099, 58929, 25]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(3161, 59050, F3, 25) (dual of [59050, 58889, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 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(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([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(3170, 59100, F3, 20) (dual of [59100, 58930, 21]-code), using Gilbert–Varšamov bound and bm = 3170 > Vbs−1(k−1) = 19 647663 364515 428307 323077 719525 878234 129146 170108 217591 012313 925788 230854 612155 [i]
- linear OA(33, 4, F3, 3) (dual of [4, 1, 4]-code or 4-arc in PG(2,3) or 4-cap in PG(2,3)), using
- dual of repetition code with length 4 [i]
- oval in PG(2, 3) [i]
- linear OA(3170, 59099, F3, 24) (dual of [59099, 58929, 25]-code), using
- construction X with Varšamov bound [i] based on
- OOA 2-folding [i] based on linear OA(3174, 59104, F3, 24) (dual of [59104, 58930, 25]-code), using
(174−24, 174, large)-Net in Base 3 — Upper bound on s
There is no (150, 174, large)-net in base 3, because
- 22 times m-reduction [i] would yield (150, 152, large)-net in base 3, but