Best Known (57, 89, s)-Nets in Base 32
(57, 89, 360)-Net over F32 — Constructive and digital
Digital (57, 89, 360)-net over F32, using
- generalized (u, u+v)-construction [i] based on
- digital (9, 19, 120)-net over F32, using
- s-reduction based on digital (9, 19, 205)-net over F32, using
- net defined by OOA [i] based on linear OOA(3219, 205, F32, 10, 10) (dual of [(205, 10), 2031, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(3219, 1025, F32, 10) (dual of [1025, 1006, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(3219, 1026, F32, 10) (dual of [1026, 1007, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(3219, 1024, F32, 10) (dual of [1024, 1005, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(3217, 1024, F32, 9) (dual of [1024, 1007, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(320, 2, F32, 0) (dual of [2, 2, 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(9) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(3219, 1026, F32, 10) (dual of [1026, 1007, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(3219, 1025, F32, 10) (dual of [1025, 1006, 11]-code), using
- net defined by OOA [i] based on linear OOA(3219, 205, F32, 10, 10) (dual of [(205, 10), 2031, 11]-NRT-code), using
- s-reduction based on digital (9, 19, 205)-net over F32, using
- digital (11, 27, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 11 and N(F) ≥ 120, using
- net from sequence [i] based on digital (11, 119)-sequence over F32, using
- digital (11, 43, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (9, 19, 120)-net over F32, using
(57, 89, 1024)-Net in Base 32 — Constructive
(57, 89, 1024)-net in base 32, using
- net defined by OOA [i] based on OOA(3289, 1024, S32, 32, 32), using
- OA 16-folding and stacking [i] based on OA(3289, 16384, S32, 32), using
- discarding factors based on OA(3289, 16386, S32, 32), using
- discarding parts of the base [i] based on linear OA(12863, 16386, F128, 32) (dual of [16386, 16323, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(30) [i] based on
- linear OA(12863, 16384, F128, 32) (dual of [16384, 16321, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(12861, 16384, F128, 31) (dual of [16384, 16323, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(1280, 2, F128, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(31) ⊂ Ce(30) [i] based on
- discarding parts of the base [i] based on linear OA(12863, 16386, F128, 32) (dual of [16386, 16323, 33]-code), using
- discarding factors based on OA(3289, 16386, S32, 32), using
- OA 16-folding and stacking [i] based on OA(3289, 16384, S32, 32), using
(57, 89, 8408)-Net over F32 — Digital
Digital (57, 89, 8408)-net over F32, using
(57, 89, large)-Net in Base 32 — Upper bound on s
There is no (57, 89, large)-net in base 32, because
- 30 times m-reduction [i] would yield (57, 59, large)-net in base 32, but