Best Known (54−14, 54, s)-Nets in Base 32
(54−14, 54, 149797)-Net over F32 — Constructive and digital
Digital (40, 54, 149797)-net over F32, using
- 321 times duplication [i] based on digital (39, 53, 149797)-net over F32, using
- net defined by OOA [i] based on linear OOA(3253, 149797, F32, 14, 14) (dual of [(149797, 14), 2097105, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(3253, 1048579, F32, 14) (dual of [1048579, 1048526, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(3253, 1048580, F32, 14) (dual of [1048580, 1048527, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(3253, 1048576, F32, 14) (dual of [1048576, 1048523, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(3249, 1048576, F32, 13) (dual of [1048576, 1048527, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(320, 4, F32, 0) (dual of [4, 4, 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(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(3253, 1048580, F32, 14) (dual of [1048580, 1048527, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(3253, 1048579, F32, 14) (dual of [1048579, 1048526, 15]-code), using
- net defined by OOA [i] based on linear OOA(3253, 149797, F32, 14, 14) (dual of [(149797, 14), 2097105, 15]-NRT-code), using
(54−14, 54, 758128)-Net over F32 — Digital
Digital (40, 54, 758128)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3254, 758128, F32, 14) (dual of [758128, 758074, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(3254, 1048585, F32, 14) (dual of [1048585, 1048531, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(3253, 1048576, F32, 14) (dual of [1048576, 1048523, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(3245, 1048576, F32, 12) (dual of [1048576, 1048531, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(321, 9, F32, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, s, F32, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(3254, 1048585, F32, 14) (dual of [1048585, 1048531, 15]-code), using
(54−14, 54, large)-Net in Base 32 — Upper bound on s
There is no (40, 54, large)-net in base 32, because
- 12 times m-reduction [i] would yield (40, 42, large)-net in base 32, but