Best Known (20, 20+9, s)-Nets in Base 49
(20, 20+9, 29462)-Net over F49 — Constructive and digital
Digital (20, 29, 29462)-net over F49, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 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
- digital (16, 25, 29412)-net over F49, using
- net defined by OOA [i] based on linear OOA(4925, 29412, F49, 9, 9) (dual of [(29412, 9), 264683, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(4925, 117649, F49, 9) (dual of [117649, 117624, 10]-code), using
- an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(4925, 117649, F49, 9) (dual of [117649, 117624, 10]-code), using
- net defined by OOA [i] based on linear OOA(4925, 29412, F49, 9, 9) (dual of [(29412, 9), 264683, 10]-NRT-code), using
- digital (0, 4, 50)-net over F49, using
(20, 20+9, 117702)-Net over F49 — Digital
Digital (20, 29, 117702)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4929, 117702, F49, 9) (dual of [117702, 117673, 10]-code), using
- (u, u+v)-construction [i] based on
- linear OA(494, 50, F49, 4) (dual of [50, 46, 5]-code or 50-arc in PG(3,49)), using
- extended Reed–Solomon code RSe(46,49) [i]
- algebraic-geometric code AG(F, Q+21P) with degQ = 3 and degPÂ =Â 2 [i] based on function field F/F49 with g(F) = 0 and N(F) ≥ 50, using the rational function field F49(x) [i]
- algebraic-geometric code AG(F,15P) with degPÂ =Â 3 [i] based on function field F/F49 with g(F) = 0 and N(F) ≥ 50 (see above)
- linear OA(4925, 117652, F49, 9) (dual of [117652, 117627, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(4925, 117649, F49, 9) (dual of [117649, 117624, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(4922, 117649, F49, 8) (dual of [117649, 117627, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(490, 3, F49, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(490, s, F49, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(494, 50, F49, 4) (dual of [50, 46, 5]-code or 50-arc in PG(3,49)), using
- (u, u+v)-construction [i] based on
(20, 20+9, large)-Net in Base 49 — Upper bound on s
There is no (20, 29, large)-net in base 49, because
- 7 times m-reduction [i] would yield (20, 22, large)-net in base 49, but