Generate Parity bits

Question

Given a binary message, and the number of parity bits, generate the associated parity bits.

A parity bit is a simple form of error detection. It's generated by counting the number of 1's in the message, if it's even attach a 0 to the end, if it's odd attach 1.
That way, if there's a 1-bit error, 3-bit error, 5-bit error, ... in the message, because of the parity-bit you know the message has been altered.
Although if there were an even number of bits altered, the parity stays the same, so you wouldn't know if the message has been changed.
Only 50% of the time you'd know if bits have been altered with one parity bit.

Generate Parity bits

To generate n parity bits for a given binary message:

1. Count the number of 1's in the message
2. Modulo by \$2^n\$
3. Attach the remainder to the message

For example, using three parity bits (n=3) and the message 10110111110110111:

1. 10110111110110111 -> 13 1's
2. \$13\mod2^3\$ -> 5
3. 10110111110110111 with 5 attached (in binary) -> 10110111110110111101

The last three digits act as parity bits.
The advantage with parity bits is, that they can't detect errors only when a multiple of \$2^n\$ bits have been altered (ignoring messages which share the same permutations). With three parity bits, you can't detect errors when 8, 16, 24, 32, 40, ... bits have been altered.
\$(1-\frac{1}{2^n})\%\$ of the time you'd know when bits have been altered, significantly more than with just one parity bit.

Rules

• Take as input a binary string or a binary array and an integer (0<n, the number of parity bits)
• Output a binary string/binary array with n parity bits attached
• The parity bits should be padded with zeros to length n
• Leading zeros are allowed
• This is code-golf, so the shortest answer wins

Test Cases

[In]:  10110, 1
[Out]: 101101
[In]:  0110101, 2
[Out]: 011010100
[In]:  1011101110, 3
[Out]: 1011101110111
[In]:  0011001100111101111010011111, 4
[Out]: 00110011001111011110100111110010

Answer 1

Retina 0.8.2, 77 bytes

,.+
,$&$*_1,$`
+`0(1*)$
$1
+`_1
11_
(1+)_+,\1*
$1
+`,(1+)\1
,$1@
@1
1
@
0
,1

Try it online! Link includes test cases. Explanation:

,.+
,$&$*_1,$`

Convert n to unary and append a copy of the message.

+`0(1*)$
$1

Delete any 0s in the copy of the message.

+`_1
11_

Calculate 2ⁿ.

(1+)_+,\1*
$1

Reduce the count of 1s modulo 2ⁿ, but add the 2ⁿ back on.

+`,(1+)\1
,$1@
@1
1

Convert the result to binary, but use @ instead of 0 to avoid disrupting the message.

@
0

Replace @s with 0s now that they're unambiguous.

,1

Remove the leading 1 of the binary number and concatenate the rest to the message.