Best Known (149, 174, s)-Nets in Base 3
(149, 174, 4927)-Net over F3 — Constructive and digital
Digital (149, 174, 4927)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 13, 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 (136, 161, 4920)-net over F3, using
- net defined by OOA [i] based on linear OOA(3161, 4920, F3, 25, 25) (dual of [(4920, 25), 122839, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3161, 59041, F3, 25) (dual of [59041, 58880, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3161, 59049, F3, 25) (dual of [59049, 58888, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(3161, 59049, F3, 25) (dual of [59049, 58888, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3161, 59041, F3, 25) (dual of [59041, 58880, 26]-code), using
- net defined by OOA [i] based on linear OOA(3161, 4920, F3, 25, 25) (dual of [(4920, 25), 122839, 26]-NRT-code), using
- digital (1, 13, 7)-net over F3, using
(149, 174, 24303)-Net over F3 — Digital
Digital (149, 174, 24303)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3174, 24303, F3, 2, 25) (dual of [(24303, 2), 48432, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3174, 29551, F3, 2, 25) (dual of [(29551, 2), 58928, 26]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3173, 29551, F3, 2, 25) (dual of [(29551, 2), 58929, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3173, 59102, F3, 25) (dual of [59102, 58929, 26]-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(312, 52, F3, 5) (dual of [52, 40, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(312, 54, F3, 5) (dual of [54, 42, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(31, 13, F3, 1) (dual of [13, 12, 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
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- linear OA(31, 13, F3, 1) (dual of [13, 12, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(312, 54, F3, 5) (dual of [54, 42, 6]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- OOA 2-folding [i] based on linear OA(3173, 59102, F3, 25) (dual of [59102, 58929, 26]-code), using
- 31 times duplication [i] based on linear OOA(3173, 29551, F3, 2, 25) (dual of [(29551, 2), 58929, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3174, 29551, F3, 2, 25) (dual of [(29551, 2), 58928, 26]-NRT-code), using
(149, 174, large)-Net in Base 3 — Upper bound on s
There is no (149, 174, large)-net in base 3, because
- 23 times m-reduction [i] would yield (149, 151, large)-net in base 3, but