Best Known (17, 27, s)-Nets in Base 7
(17, 27, 121)-Net over F7 — Constructive and digital
Digital (17, 27, 121)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (2, 7, 21)-net over F7, using
- net defined by OOA [i] based on linear OOA(77, 21, F7, 5, 5) (dual of [(21, 5), 98, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(77, 43, F7, 5) (dual of [43, 36, 6]-code), using
- net defined by OOA [i] based on linear OOA(77, 21, F7, 5, 5) (dual of [(21, 5), 98, 6]-NRT-code), using
- digital (10, 20, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 10, 50)-net over F49, using
- net from sequence [i] based on digital (0, 49)-sequence over F49, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 0 and N(F) ≥ 50, using
- the rational function field F49(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 49)-sequence over F49, using
- trace code for nets [i] based on digital (0, 10, 50)-net over F49, using
- digital (2, 7, 21)-net over F7, using
(17, 27, 346)-Net over F7 — Digital
Digital (17, 27, 346)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(727, 346, F7, 10) (dual of [346, 319, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(727, 353, F7, 10) (dual of [353, 326, 11]-code), using
- construction XX applied to C1 = C([49,57]), C2 = C([52,58]), C3 = C1 + C2 = C([52,57]), and C∩ = C1 ∩ C2 = C([49,58]) [i] based on
- linear OA(722, 342, F7, 9) (dual of [342, 320, 10]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {49,50,…,57}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(719, 342, F7, 7) (dual of [342, 323, 8]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {52,53,…,58}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(725, 342, F7, 10) (dual of [342, 317, 11]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {49,50,…,58}, and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(716, 342, F7, 6) (dual of [342, 326, 7]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {52,53,…,57}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(72, 8, F7, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,7)), using
- extended Reed–Solomon code RSe(6,7) [i]
- Hamming code H(2,7) [i]
- algebraic-geometric code AG(F, Q+1P) with degQ = 3 and degPÂ =Â 2 [i] based on function field F/F7 with g(F) = 0 and N(F) ≥ 8, using the rational function field F7(x) [i]
- linear OA(70, 3, F7, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction XX applied to C1 = C([49,57]), C2 = C([52,58]), C3 = C1 + C2 = C([52,57]), and C∩ = C1 ∩ C2 = C([49,58]) [i] based on
- discarding factors / shortening the dual code based on linear OA(727, 353, F7, 10) (dual of [353, 326, 11]-code), using
(17, 27, 15890)-Net in Base 7 — Upper bound on s
There is no (17, 27, 15891)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 65723 132327 763378 452635 > 727 [i]