Best Known (49−12, 49, s)-Nets in Base 32
(49−12, 49, 174766)-Net over F32 — Constructive and digital
Digital (37, 49, 174766)-net over F32, using
- net defined by OOA [i] based on linear OOA(3249, 174766, F32, 12, 12) (dual of [(174766, 12), 2097143, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(3249, 1048596, F32, 12) (dual of [1048596, 1048547, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(3249, 1048600, F32, 12) (dual of [1048600, 1048551, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(6) [i] based on
- 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(3225, 1048576, F32, 7) (dual of [1048576, 1048551, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(324, 24, F32, 4) (dual of [24, 20, 5]-code or 24-arc in PG(3,32)), using
- discarding factors / shortening the dual code based on linear OA(324, 32, F32, 4) (dual of [32, 28, 5]-code or 32-arc in PG(3,32)), using
- Reed–Solomon code RS(28,32) [i]
- discarding factors / shortening the dual code based on linear OA(324, 32, F32, 4) (dual of [32, 28, 5]-code or 32-arc in PG(3,32)), using
- construction X applied to Ce(11) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(3249, 1048600, F32, 12) (dual of [1048600, 1048551, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(3249, 1048596, F32, 12) (dual of [1048596, 1048547, 13]-code), using
(49−12, 49, 349526)-Net in Base 32 — Constructive
(37, 49, 349526)-net in base 32, using
- base change [i] based on digital (23, 35, 349526)-net over F128, using
- net defined by OOA [i] based on linear OOA(12835, 349526, F128, 12, 12) (dual of [(349526, 12), 4194277, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(12835, 2097156, F128, 12) (dual of [2097156, 2097121, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(12835, 2097159, F128, 12) (dual of [2097159, 2097124, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(12834, 2097152, F128, 12) (dual of [2097152, 2097118, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(12828, 2097152, F128, 10) (dual of [2097152, 2097124, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(1281, 7, F128, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(12835, 2097159, F128, 12) (dual of [2097159, 2097124, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(12835, 2097156, F128, 12) (dual of [2097156, 2097121, 13]-code), using
- net defined by OOA [i] based on linear OOA(12835, 349526, F128, 12, 12) (dual of [(349526, 12), 4194277, 13]-NRT-code), using
(49−12, 49, 1048600)-Net over F32 — Digital
Digital (37, 49, 1048600)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3249, 1048600, F32, 12) (dual of [1048600, 1048551, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(6) [i] based on
- 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(3225, 1048576, F32, 7) (dual of [1048576, 1048551, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(324, 24, F32, 4) (dual of [24, 20, 5]-code or 24-arc in PG(3,32)), using
- discarding factors / shortening the dual code based on linear OA(324, 32, F32, 4) (dual of [32, 28, 5]-code or 32-arc in PG(3,32)), using
- Reed–Solomon code RS(28,32) [i]
- discarding factors / shortening the dual code based on linear OA(324, 32, F32, 4) (dual of [32, 28, 5]-code or 32-arc in PG(3,32)), using
- construction X applied to Ce(11) ⊂ Ce(6) [i] based on
(49−12, 49, large)-Net in Base 32 — Upper bound on s
There is no (37, 49, large)-net in base 32, because
- 10 times m-reduction [i] would yield (37, 39, large)-net in base 32, but