Best Known (46−17, 46, s)-Nets in Base 7
(46−17, 46, 116)-Net over F7 — Constructive and digital
Digital (29, 46, 116)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (4, 12, 16)-net over F7, using
- 4 times m-reduction [i] based on digital (4, 16, 16)-net over F7, using
- digital (17, 34, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 17, 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, 17, 50)-net over F49, using
- digital (4, 12, 16)-net over F7, using
(46−17, 46, 359)-Net over F7 — Digital
Digital (29, 46, 359)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(746, 359, F7, 17) (dual of [359, 313, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(746, 360, F7, 17) (dual of [360, 314, 18]-code), using
- 9 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 6 times 0) [i] based on linear OA(743, 348, F7, 17) (dual of [348, 305, 18]-code), using
- construction XX applied to C1 = C([341,14]), C2 = C([0,15]), C3 = C1 + C2 = C([0,14]), and C∩ = C1 ∩ C2 = C([341,15]) [i] based on
- linear OA(740, 342, F7, 16) (dual of [342, 302, 17]-code), using the primitive BCH-code C(I) with length 342 = 73−1, defining interval I = {−1,0,…,14}, and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(740, 342, F7, 16) (dual of [342, 302, 17]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 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 = {−1,0,…,15}, and designed minimum distance d ≥ |I|+1 = 18 [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(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
- linear OA(70, 3, F7, 0) (dual of [3, 3, 1]-code) (see above)
- construction XX applied to C1 = C([341,14]), C2 = C([0,15]), C3 = C1 + C2 = C([0,14]), and C∩ = C1 ∩ C2 = C([341,15]) [i] based on
- 9 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 6 times 0) [i] based on linear OA(743, 348, F7, 17) (dual of [348, 305, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(746, 360, F7, 17) (dual of [360, 314, 18]-code), using
(46−17, 46, 35575)-Net in Base 7 — Upper bound on s
There is no (29, 46, 35576)-net in base 7, because
- 1 times m-reduction [i] would yield (29, 45, 35576)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 107 008525 734669 381433 210489 346981 521921 > 745 [i]