Best Known (60, 60+13, s)-Nets in Base 8
(60, 60+13, 43700)-Net over F8 — Constructive and digital
Digital (60, 73, 43700)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (0, 6, 9)-net over F8, using
- net from sequence [i] based on digital (0, 8)-sequence over F8, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 0 and N(F) ≥ 9, using
- the rational function field F8(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 8)-sequence over F8, using
- digital (54, 67, 43691)-net over F8, using
- net defined by OOA [i] based on linear OOA(867, 43691, F8, 13, 13) (dual of [(43691, 13), 567916, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(867, 262147, F8, 13) (dual of [262147, 262080, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(867, 262150, F8, 13) (dual of [262150, 262083, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(867, 262144, F8, 13) (dual of [262144, 262077, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(861, 262144, F8, 12) (dual of [262144, 262083, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(80, 6, F8, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(867, 262150, F8, 13) (dual of [262150, 262083, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(867, 262147, F8, 13) (dual of [262147, 262080, 14]-code), using
- net defined by OOA [i] based on linear OOA(867, 43691, F8, 13, 13) (dual of [(43691, 13), 567916, 14]-NRT-code), using
- digital (0, 6, 9)-net over F8, using
(60, 60+13, 262176)-Net over F8 — Digital
Digital (60, 73, 262176)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(873, 262176, F8, 13) (dual of [262176, 262103, 14]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(871, 262172, F8, 13) (dual of [262172, 262101, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- linear OA(867, 262144, F8, 13) (dual of [262144, 262077, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(843, 262144, F8, 9) (dual of [262144, 262101, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(84, 28, F8, 3) (dual of [28, 24, 4]-code or 28-cap in PG(3,8)), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- linear OA(871, 262174, F8, 12) (dual of [262174, 262103, 13]-code), using Gilbert–Varšamov bound and bm = 871 > Vbs−1(k−1) = 19 920572 510806 350026 034127 754669 329283 173501 845374 452963 356356 [i]
- linear OA(80, 2, F8, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(871, 262172, F8, 13) (dual of [262172, 262101, 14]-code), using
- construction X with Varšamov bound [i] based on
(60, 60+13, large)-Net in Base 8 — Upper bound on s
There is no (60, 73, large)-net in base 8, because
- 11 times m-reduction [i] would yield (60, 62, large)-net in base 8, but