Best Known (70−18, 70, s)-Nets in Base 32
(70−18, 70, 116509)-Net over F32 — Constructive and digital
Digital (52, 70, 116509)-net over F32, using
- net defined by OOA [i] based on linear OOA(3270, 116509, F32, 18, 18) (dual of [(116509, 18), 2097092, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(3270, 1048581, F32, 18) (dual of [1048581, 1048511, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3270, 1048585, F32, 18) (dual of [1048585, 1048515, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- linear OA(3269, 1048576, F32, 18) (dual of [1048576, 1048507, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(3261, 1048576, F32, 16) (dual of [1048576, 1048515, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(17) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(3270, 1048585, F32, 18) (dual of [1048585, 1048515, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(3270, 1048581, F32, 18) (dual of [1048581, 1048511, 19]-code), using
(70−18, 70, 679409)-Net over F32 — Digital
Digital (52, 70, 679409)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3270, 679409, F32, 18) (dual of [679409, 679339, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3270, 1048585, F32, 18) (dual of [1048585, 1048515, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- linear OA(3269, 1048576, F32, 18) (dual of [1048576, 1048507, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(3261, 1048576, F32, 16) (dual of [1048576, 1048515, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(17) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(3270, 1048585, F32, 18) (dual of [1048585, 1048515, 19]-code), using
(70−18, 70, large)-Net in Base 32 — Upper bound on s
There is no (52, 70, large)-net in base 32, because
- 16 times m-reduction [i] would yield (52, 54, large)-net in base 32, but