Best Known (32, 51, s)-Nets in Base 7
(32, 51, 116)-Net over F7 — Constructive and digital
Digital (32, 51, 116)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (4, 13, 16)-net over F7, using
- 3 times m-reduction [i] based on digital (4, 16, 16)-net over F7, using
- digital (19, 38, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 19, 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, 19, 50)-net over F49, using
- digital (4, 13, 16)-net over F7, using
(32, 51, 356)-Net over F7 — Digital
Digital (32, 51, 356)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(751, 356, F7, 19) (dual of [356, 305, 20]-code), using
- construction XX applied to C1 = C([340,14]), C2 = C([0,16]), C3 = C1 + C2 = C([0,14]), and C∩ = C1 ∩ C2 = C([340,16]) [i] based on
- linear OA(743, 342, F7, 17) (dual of [342, 299, 18]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {−2,−1,…,14}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(743, 342, F7, 17) (dual of [342, 299, 18]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [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 = {−2,−1,…,16}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(737, 342, F7, 15) (dual of [342, 305, 16]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(71, 7, F7, 1) (dual of [7, 6, 2]-code), using
- Reed–Solomon code RS(6,7) [i]
- linear OA(71, 7, F7, 1) (dual of [7, 6, 2]-code) (see above)
- construction XX applied to C1 = C([340,14]), C2 = C([0,16]), C3 = C1 + C2 = C([0,14]), and C∩ = C1 ∩ C2 = C([340,16]) [i] based on
(32, 51, 34238)-Net in Base 7 — Upper bound on s
There is no (32, 51, 34239)-net in base 7, because
- 1 times m-reduction [i] would yield (32, 50, 34239)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 1 798521 573948 678574 067616 494633 078023 725643 > 750 [i]