Best Known (167, 167+27, s)-Nets in Base 3
(167, 167+27, 4549)-Net over F3 — Constructive and digital
Digital (167, 194, 4549)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 14, 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 (153, 180, 4542)-net over F3, using
- net defined by OOA [i] based on linear OOA(3180, 4542, F3, 27, 27) (dual of [(4542, 27), 122454, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3180, 59047, F3, 27) (dual of [59047, 58867, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3180, 59048, F3, 27) (dual of [59048, 58868, 28]-code), using
- 1 times truncation [i] based on linear OA(3181, 59049, F3, 28) (dual of [59049, 58868, 29]-code), using
- an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- 1 times truncation [i] based on linear OA(3181, 59049, F3, 28) (dual of [59049, 58868, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3180, 59048, F3, 27) (dual of [59048, 58868, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3180, 59047, F3, 27) (dual of [59047, 58867, 28]-code), using
- net defined by OOA [i] based on linear OOA(3180, 4542, F3, 27, 27) (dual of [(4542, 27), 122454, 28]-NRT-code), using
- digital (1, 14, 7)-net over F3, using
(167, 167+27, 29551)-Net over F3 — Digital
Digital (167, 194, 29551)-net over F3, using
- 31 times duplication [i] based on digital (166, 193, 29551)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3193, 29551, F3, 2, 27) (dual of [(29551, 2), 58909, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3193, 59102, F3, 27) (dual of [59102, 58909, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,10]) [i] based on
- linear OA(3181, 59050, F3, 27) (dual of [59050, 58869, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(3141, 59050, F3, 21) (dual of [59050, 58909, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (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([0,13]) ⊂ C([0,10]) [i] based on
- OOA 2-folding [i] based on linear OA(3193, 59102, F3, 27) (dual of [59102, 58909, 28]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3193, 29551, F3, 2, 27) (dual of [(29551, 2), 58909, 28]-NRT-code), using
(167, 167+27, large)-Net in Base 3 — Upper bound on s
There is no (167, 194, large)-net in base 3, because
- 25 times m-reduction [i] would yield (167, 169, large)-net in base 3, but