Best Known (50, 69, s)-Nets in Base 32
(50, 69, 3718)-Net over F32 — Constructive and digital
Digital (50, 69, 3718)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (5, 14, 77)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 0 and N(F) ≥ 33, using
- the rational function field F32(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 32)-sequence over F32, using
- digital (1, 10, 44)-net over F32, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 1 and N(F) ≥ 44, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- digital (0, 4, 33)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (36, 55, 3641)-net over F32, using
- net defined by OOA [i] based on linear OOA(3255, 3641, F32, 19, 19) (dual of [(3641, 19), 69124, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3255, 32770, F32, 19) (dual of [32770, 32715, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3255, 32771, F32, 19) (dual of [32771, 32716, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(3255, 32768, F32, 19) (dual of [32768, 32713, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3252, 32768, F32, 18) (dual of [32768, 32716, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(320, 3, F32, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(320, s, F32, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(3255, 32771, F32, 19) (dual of [32771, 32716, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3255, 32770, F32, 19) (dual of [32770, 32715, 20]-code), using
- net defined by OOA [i] based on linear OOA(3255, 3641, F32, 19, 19) (dual of [(3641, 19), 69124, 20]-NRT-code), using
- digital (5, 14, 77)-net over F32, using
(50, 69, 29128)-Net in Base 32 — Constructive
(50, 69, 29128)-net in base 32, using
- net defined by OOA [i] based on OOA(3269, 29128, S32, 19, 19), using
- OOA 9-folding and stacking with additional row [i] based on OA(3269, 262153, S32, 19), using
- discarding factors based on OA(3269, 262155, S32, 19), using
- discarding parts of the base [i] based on linear OA(6457, 262155, F64, 19) (dual of [262155, 262098, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(6455, 262144, F64, 19) (dual of [262144, 262089, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(6446, 262144, F64, 16) (dual of [262144, 262098, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(642, 11, F64, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,64)), using
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- Reed–Solomon code RS(62,64) [i]
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- discarding parts of the base [i] based on linear OA(6457, 262155, F64, 19) (dual of [262155, 262098, 20]-code), using
- discarding factors based on OA(3269, 262155, S32, 19), using
- OOA 9-folding and stacking with additional row [i] based on OA(3269, 262153, S32, 19), using
(50, 69, 143396)-Net over F32 — Digital
Digital (50, 69, 143396)-net over F32, using
(50, 69, large)-Net in Base 32 — Upper bound on s
There is no (50, 69, large)-net in base 32, because
- 17 times m-reduction [i] would yield (50, 52, large)-net in base 32, but