Best Known (17−6, 17, s)-Nets in Base 3
(17−6, 17, 53)-Net over F3 — Constructive and digital
Digital (11, 17, 53)-net over F3, using
- net defined by OOA [i] based on linear OOA(317, 53, F3, 6, 6) (dual of [(53, 6), 301, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(317, 53, F3, 5, 6) (dual of [(53, 5), 248, 7]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(35, 20, F3, 5, 3) (dual of [(20, 5), 95, 4]-NRT-code), using
- appending 2 arbitrary columns [i] based on linear OOA(35, 20, F3, 3, 3) (dual of [(20, 3), 55, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(35, 20, F3, 2, 3) (dual of [(20, 2), 35, 4]-NRT-code), using
- appending 2 arbitrary columns [i] based on linear OOA(35, 20, F3, 3, 3) (dual of [(20, 3), 55, 4]-NRT-code), using
- linear OOA(312, 33, F3, 5, 6) (dual of [(33, 5), 153, 7]-NRT-code), using
- extracting embedded OOA [i] based on digital (6, 12, 33)-net over F3, using
- linear OOA(35, 20, F3, 5, 3) (dual of [(20, 5), 95, 4]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(317, 53, F3, 5, 6) (dual of [(53, 5), 248, 7]-NRT-code), using
(17−6, 17, 87)-Net over F3 — Digital
Digital (11, 17, 87)-net over F3, using
- net defined by OOA [i] based on linear OOA(317, 87, F3, 6, 6) (dual of [(87, 6), 505, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(317, 87, F3, 5, 6) (dual of [(87, 5), 418, 7]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(317, 87, F3, 6) (dual of [87, 70, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(317, 88, F3, 6) (dual of [88, 71, 7]-code), using
- construction XX applied to C1 = C({0,1,2,53}), C2 = C([0,4]), C3 = C1 + C2 = C([0,2]), and C∩ = C1 ∩ C2 = C({0,1,2,4,53}) [i] based on
- linear OA(313, 80, F3, 5) (dual of [80, 67, 6]-code), using the primitive cyclic code C(A) with length 80 = 34−1, defining set A = {0,1,2,53}, and minimum distance d ≥ |{−1,0,1,2,3}|+1 = 6 (BCH-bound) [i]
- linear OA(313, 80, F3, 5) (dual of [80, 67, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(317, 80, F3, 6) (dual of [80, 63, 7]-code), using the primitive cyclic code C(A) with length 80 = 34−1, defining set A = {0,1,2,4,53}, and minimum distance d ≥ |{−1,0,…,4}|+1 = 7 (BCH-bound) [i]
- 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]
- linear OA(30, 4, F3, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(30, 4, F3, 0) (dual of [4, 4, 1]-code) (see above)
- construction XX applied to C1 = C({0,1,2,53}), C2 = C([0,4]), C3 = C1 + C2 = C([0,2]), and C∩ = C1 ∩ C2 = C({0,1,2,4,53}) [i] based on
- discarding factors / shortening the dual code based on linear OA(317, 88, F3, 6) (dual of [88, 71, 7]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(317, 87, F3, 6) (dual of [87, 70, 7]-code), using
- appending kth column [i] based on linear OOA(317, 87, F3, 5, 6) (dual of [(87, 5), 418, 7]-NRT-code), using
(17−6, 17, 456)-Net in Base 3 — Upper bound on s
There is no (11, 17, 457)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 129 353851 > 317 [i]