Best Known (39, 39+24, s)-Nets in Base 7
(39, 39+24, 115)-Net over F7 — Constructive and digital
Digital (39, 63, 115)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (1, 13, 13)-net over F7, using
- digital (26, 50, 102)-net over F7, using
- trace code for nets [i] based on digital (1, 25, 51)-net over F49, using
- net from sequence [i] based on digital (1, 50)-sequence over F49, using
- trace code for nets [i] based on digital (1, 25, 51)-net over F49, using
(39, 39+24, 351)-Net over F7 — Digital
Digital (39, 63, 351)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(763, 351, F7, 24) (dual of [351, 288, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(763, 353, F7, 24) (dual of [353, 290, 25]-code), using
- construction XX applied to C1 = C([35,57]), C2 = C([38,58]), C3 = C1 + C2 = C([38,57]), and C∩ = C1 ∩ C2 = C([35,58]) [i] based on
- linear OA(758, 342, F7, 23) (dual of [342, 284, 24]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {35,36,…,57}, and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(755, 342, F7, 21) (dual of [342, 287, 22]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {38,39,…,58}, and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(761, 342, F7, 24) (dual of [342, 281, 25]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {35,36,…,58}, and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(752, 342, F7, 20) (dual of [342, 290, 21]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {38,39,…,57}, and designed minimum distance d ≥ |I|+1 = 21 [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([35,57]), C2 = C([38,58]), C3 = C1 + C2 = C([38,57]), and C∩ = C1 ∩ C2 = C([35,58]) [i] based on
- discarding factors / shortening the dual code based on linear OA(763, 353, F7, 24) (dual of [353, 290, 25]-code), using
(39, 39+24, 24090)-Net in Base 7 — Upper bound on s
There is no (39, 63, 24091)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 174324 329469 730946 630933 407257 058657 709142 054582 161881 > 763 [i]