Best Known (58−22, 58, s)-Nets in Base 7
(58−22, 58, 115)-Net over F7 — Constructive and digital
Digital (36, 58, 115)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (1, 12, 13)-net over F7, using
- 1 times m-reduction [i] based on digital (1, 13, 13)-net over F7, using
- digital (24, 46, 102)-net over F7, using
- trace code for nets [i] based on digital (1, 23, 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, 23, 51)-net over F49, using
- digital (1, 12, 13)-net over F7, using
(58−22, 58, 344)-Net over F7 — Digital
Digital (36, 58, 344)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(758, 344, F7, 22) (dual of [344, 286, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(758, 354, F7, 22) (dual of [354, 296, 23]-code), using
- construction XX applied to C1 = C([45,64]), C2 = C([43,61]), C3 = C1 + C2 = C([45,61]), and C∩ = C1 ∩ C2 = C([43,64]) [i] based on
- 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 = {45,46,…,64}, and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(749, 342, F7, 19) (dual of [342, 293, 20]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {43,44,…,61}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(755, 342, F7, 22) (dual of [342, 287, 23]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {43,44,…,64}, and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(746, 342, F7, 17) (dual of [342, 296, 18]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {45,46,…,61}, and designed minimum distance d ≥ |I|+1 = 18 [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(71, 4, F7, 1) (dual of [4, 3, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, 7, F7, 1) (dual of [7, 6, 2]-code), using
- Reed–Solomon code RS(6,7) [i]
- discarding factors / shortening the dual code based on linear OA(71, 7, F7, 1) (dual of [7, 6, 2]-code), using
- construction XX applied to C1 = C([45,64]), C2 = C([43,61]), C3 = C1 + C2 = C([45,61]), and C∩ = C1 ∩ C2 = C([43,64]) [i] based on
- discarding factors / shortening the dual code based on linear OA(758, 354, F7, 22) (dual of [354, 296, 23]-code), using
(58−22, 58, 23372)-Net in Base 7 — Upper bound on s
There is no (36, 58, 23373)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 10 369633 782573 732273 425628 368462 445692 192145 260599 > 758 [i]